Knowledge of Galois Fields is must for understanding Error Control Codes. This presentation undertakes concepts of Galois Field required for understanding Error Control Codes in very simple manner, explaining its complex mathematical intricacies in a structured manner.
Kinds of Propagation Models
Models of Different Types of Cells
Web Plot Digitizer Tool
Study of the parameters fc, d, hb, hm and Coverage Environments for each of OKUMURA, HATA and COST231
MATLAB Simulation
Kinds of Propagation Models
Models of Different Types of Cells
Web Plot Digitizer Tool
Study of the parameters fc, d, hb, hm and Coverage Environments for each of OKUMURA, HATA and COST231
MATLAB Simulation
An antenna array (or array antenna) is a set of multiple connected antennas which work together as a single antenna, to transmit or receive radio waves. The individual antenna elements are connected to a single receiver or transmitter by feedlines that feed the power to the elements in a specific phase relationship. The radio waves radiated by each individual antenna combine and superpose, adding together (interfering constructively) to enhance the power radiated in desired directions, and cancelling (interfering destructively) to reduce the power radiated in other directions. Similarly, when used for receiving, the separate radio frequency currents from the individual antennas combine in the receiver with the correct phase relationship to enhance signals received from the desired directions and cancel signals from undesired directions.
Error control codes are necessary for transmission and storage of large volumes of date sensitive to errors. BCH codes and Reed Solomon codes are the most important class of multiple error correcting codes for binary and non-binary channels respectively. Peterson and later Berlekamp and Massey discovered powerful algorithms which became viable with the help of new digital technology. Use of Galois fields gave a structured approach to designing of these codes. This presentation deals with above in a very structured and systematic manner.
Spread spectrum is a communication technique that spreads a narrowband communication signal over a wide range of frequencies for transmission then de-spreads it into the original data bandwidth at the receive.
–concept of groups, rings, fields
–modular arithmetic with integers
–Euclid’s algorithm for GCD
–finite fields GF(p)
–polynomial arithmetic in general and in GF(2n)
An antenna array (or array antenna) is a set of multiple connected antennas which work together as a single antenna, to transmit or receive radio waves. The individual antenna elements are connected to a single receiver or transmitter by feedlines that feed the power to the elements in a specific phase relationship. The radio waves radiated by each individual antenna combine and superpose, adding together (interfering constructively) to enhance the power radiated in desired directions, and cancelling (interfering destructively) to reduce the power radiated in other directions. Similarly, when used for receiving, the separate radio frequency currents from the individual antennas combine in the receiver with the correct phase relationship to enhance signals received from the desired directions and cancel signals from undesired directions.
Error control codes are necessary for transmission and storage of large volumes of date sensitive to errors. BCH codes and Reed Solomon codes are the most important class of multiple error correcting codes for binary and non-binary channels respectively. Peterson and later Berlekamp and Massey discovered powerful algorithms which became viable with the help of new digital technology. Use of Galois fields gave a structured approach to designing of these codes. This presentation deals with above in a very structured and systematic manner.
Spread spectrum is a communication technique that spreads a narrowband communication signal over a wide range of frequencies for transmission then de-spreads it into the original data bandwidth at the receive.
–concept of groups, rings, fields
–modular arithmetic with integers
–Euclid’s algorithm for GCD
–finite fields GF(p)
–polynomial arithmetic in general and in GF(2n)
Viterbi decoding algorithm is a complete decoding algorithm with zero probability of decoding failure, but larger probability of decoding error than an incomplete decoder. It is practical for binary codes of small constraint length. Stack algorithm reduces computational work by keeping track of paths it has already traversed...
Preference of Efficient Architectures for GF(p) Elliptic Curve Crypto Operati...CSCJournals
This paper explores architecture possibilities to utilize more than one multiplier to speedup the computation of GF(p) elliptic curve crypto systems. The architectures considers projective coordinates to reduce the GF(p) inversion complexity through additional multiplication operations. The study compares the standard projective coordinates (X/Z,Y/Z) with the Jacobian coordinates (X/Z2,Y/Z3) exploiting their multiplication operations parallelism. We assume using 2, 3, 4, and 5 parallel multipliers and accordingly choose the appropriate projective coordinate efficiently. The study proved that the Jacobian coordinates (X/Z2,Y/Z3) is preferred when single or two multipliers are used. Whenever 3 or 4 multipliers are available, the standard projective coordinates (X/Z,Y/Z) are favored. We found that designs with 5 multipliers have no benefit over the 4 multipliers because of the data dependency. These architectures study are particularly attractive for elliptic curve cryptosystems when hardware area optimization is the key concern.
Semantic Compositionality through Recursive Matrix-Vector Spaces (Socher et al.)marujirou
Slides about introducing a paper "Socher et al. Semantic Compositionality through Recursive Matrix-Vector Spaces" presented at DL reading group at Tokyo Metropolitan University, Japan.
Please download, and use Slideshow mode to view the slides, as that's where the magic of animation moves the telemetry frame elements around to illustrate the inner workings.
JPL / NASA Deep Space Network Telemetry
file ini berisikan persentasi tentang salah satu teknik pengkodean yang digunakan untuk deteksi dan koreksi kesalahan pada pengiriman data yang disebut dengan reed solomon code.
BCH codes, part of the cyclic codes, are very powerful error correcting codes widely used in the information coding techniques. This presentation explains these codes with an example.
This presentation enables users to understand basics of Information Theory, Entropy, Binary channels, channel capacity and error condition in easy and detailed manner. Concepts are explained properly using derivations and examples.
Embark on a fascinating exploration of Group Theory in Discrete Mathematics through this enlightening PowerPoint presentation. This presentation delves into the foundational concepts of groups, providing a clear understanding of their structure, properties, and applications in various mathematical disciplines.
cryptography slides. it consists of all the lecture notes of ankur sodhi. students of lpu final year btech computer sc. can take it as a reference if needed
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly.
INTRODUCTION: Fibre optical sensors offer number of distinct advantages which makes them unique for many applications where conventional sensors are difficult or impossible to deploy or can not provide the same wealth of information. They are completely passive, hence can be used in explosive environment. Immunity to electromagnetic interference makes it ideal for microwave environment. They are resistant to high temperatures and chemically reactive environment, ideal for harsh and hostile environment. Small size makes it ideal for embedding and surface mounting. Has high degree of biocompatibility, non-intrusive nature and electromagnetic immunity, ideal for medical applications like intra-aortic balloon pumping. They can monitor a wide range of physical and chemical parameters. It has potential for very high sensitivity, range and resolution. Complete electrical insulation from high electrostatic potential and Remote operation over several km lengths without any lead sensitivity makes it ideal for deployment in boreholes or measurements in hazardous environment. Unique multiplexed and distributed sensors provide measurements at large number of points along single optical cable, ideal for minimising cable deployment and cable weight, monitoring extended structures like pipelines, dams.
Various types of sensors are Point sensors, Integrated Sensors, Quasidistributed multiplexed sensors, Distributed sensors. Examples of such sensors are Fabry-Perot sensors, Single Fibre Bragg Grating sensors, Integrated strain sensor, Intruder Pressure sensor, Strain/Force sensor, Position sensor, Temperature sensor, Deformation sensor etc.
Orthogonal Frequency Division Multiplexing, OFDM uses a large number of narrow sub-carriers for multi-carrier transmission to overcome the effect of multi path fading problem. LTE uses OFDM for the downlink, from base station to terminal to transmit the data over many narrow band careers of 180 KHz each instead of spreading one signal over the complete 5MHz career bandwidth. OFDM meets the LTE requirement for spectrum flexibility and enables cost-efficient solutions for very wide carriers with high peak rates.
The primary advantage of OFDM over single-carrier schemes is its ability to cope with severe channel conditions. Channel equalization is simplified. The low symbol rate makes the use of a guard interval between symbols affordable, making it possible to eliminate inter symbol interference (ISI).
Small cells are Low-powered radio access nodes, Operate in licensed and unlicensed spectrum, Short range mobile phone base stations, Range from very compact residential femto-cells of area 10 meters to larger equipment used inside commercial offices or outdoor public spaces of area 1 or 2 kilometers, "small" compared to a mobile macro cell, with range of a few tens of kilometers, Complements mobile phone service from larger macro cell towers, Offer excellent mobile phone coverage and data speeds at home, in the office and public areas for both voice and data, Developed for both 3G and the newer 4G/LTE radio technologies.
Femto cells are Initially designed for residential and small business use with a short range and a limited number of channels. Femtocell devices use licenced radio spectrum. Femto cells must be operated and controlled by a mobile phone company, One cell with one mobile phone operator. When in range, the mobile phone will detect cell and use it in preference to the larger macrocell sites. Calls are made and received in exactly the same way as macrocell. Except, the signals are sent encrypted from the small cell via the public or private broadband IP network to one of the mobile operators main switching centres.
Light Fidelity (Li-Fi) is a bidirectional, high speed , fully networked wireless communication technology similar to Wi-Fi. Li-Fi was first put forward by Professor Harald Haas,University of Edinburgh, during a TED Talk in 2011. Li-Fi is a form of visible light communication and a subset of optical wireless communications (OWC) and could be a complement to RF communication (Wi-Fi or Cellular network), or even a replacement in contexts of data broadcasting. It is so far measured to be about 100 times faster than some Wi-Fi implementations, reaching speeds of 224 gigabits per second.
Optical Fiber Communication Part 3 Optical Digital ReceiverMadhumita Tamhane
Current generated by photodetector is very weak and is adversely effected by random noises associated with photo detection process. When amplified, this signal further gets corrupted by amplifiers. Noise considerations are thus important in designing optical receivers.
Most meaningful criteria for measuring performance of a digital communication system is average error probability, and in analog system, it is peak signal to rms noise ratio. ...
Optical fiber communication Part 2 Sources and DetectorsMadhumita Tamhane
For optical fiber communication, major light sources are hetero-junction-structured semiconductor laser diode and light emitting diodes. Heterojunction consists of two adjoining semiconductor materials with different bandgap energies. They have adequate power for wide range of applications. Detectors used are PiN diode and Avalanche Photodiode. Being very small in size and feeding to small core optical fiber, it is very important to study emission characteristics of sources and their coupling to fiber. As it can operate for low power over a long distance, received power is very small, hence study of noise characteristics of detectors is very essential...
Optical fiber communication Part 1 Optical Fiber FundamentalsMadhumita Tamhane
Optical fiber systems grew from combination of semiconductor technology, which provided necessary light sources and photodetectors and optical waveguide technology. It has significant inherent advantages over conventional copper systems- low transmission loss, wide BW, light weight and size, immunity to interferences, signal security to name a few. One principle characteristic of optical fiber is its attenuation as a function of wavelength. Hence it is operated in two major low attenuation wavelength windows 800-900nm and 1100-1600nm . Light travels inside optical fiber waveguide on principle of total internal reflection. Fiber is available as single mode and multiple mode, step index and graded index depending on applications and expenditures. Principle of fiber can be understood by ray theory or mode theory. ...
Main constraint for colour TV was compatibility with existing monochrome system. It should produce normal black and white picture on monochrome receiver without any modification on receiver circuitry. Moreover colour receiver must produce a black and white picture if transmission is monochrome.
Hence it should have same - bandwidth, location & spacing of sound and video frequencies, luminance information as a monochromatic signal. Colour information in signal should not effect picture on a monochrome receiver. Other circuit details of colour receiver should be same as that of monochromatic receiver..
Fundamental aim of Television is to extend the sense of sight beyond its natural limits, along with associated sound. It is radio communication of sound along with picture details. The picture signal is amplitude modulated sound signal frequency modulated before transmission. Carrier frequencies are suitably spaced so that combined signal can be radiated through a common antenna. Each broadcasting station can have its own carrier frequency and receiver can be tuned to select desired stations by tuning to respective frequency...
Field of telecommunications has evolved from crudest form of communications to electrical, radio and electro-optical communications. From manual exchange like local battery, central battery exchange, to crossbar switching, director system and to common control systems, telephone communications had started evolving to cater to better and better specifications and needs. Touch tone dial telephone opened a new horizon in the field of end to end signalling. Then came computerised stored program control systems, various multiplexing techniques. With increase in traffic there was a need to study traffic and blocking capabilities....
X.25 is a packet-switched network, developed by ITU-T as an interface between data terminal equipment DTE and data circuit-terminating equipment DCE for terminal operation in packet mode on public data network. It is an end-to-end protocol, but actual movement of packet through the network is invisible to the user.The user sees the network as a cloud through which each packet passes on its way to the receiving DTE.
It defines how a packet-mode terminal can be connected to a packet network for exchange of data. It describes procedures necessary for establishing, maintaining and terminating connections. It uses virtual network approach to packet switching, SVC and PVC and uses asynchronous TDM to multiplex data...
SDLC is synchronous bit oriented protocol developed by IBM for serial-by-bit information transfer over a data communication channel. Using EBCDIC, data is transferred in frames. Primary station controls data transfer and issues command while secondary station receives command responses to primary.
HDLC is superset of SDLC. Hence, it gives added facilities of extended addressing, CRC-16, extended control field, where 127 frames can be sent together without receiving an acknowledgement. It also allows balance mode of operation analogous to point to point communication...
Data communication protocols in centralised networks (in master:slave environ...Madhumita Tamhane
Data communication protocols can be asynchronous or synchronous handling respective data formats. Asynchronous protocols are character oriented while synchronous protocols can be either character oriented or bit oriented protocols. Most commonly used Asynchronous character oriented protocol is IBM's asynchronous data link protocol 83B. Most commonly used Synchronous character oriented protocol is IBM's BISYNC which is addressed in this presentation while most common bit oriented portals are SDLC and HDLC which are addressed in next presentation.
The line control unit LCU has several important functions. LCU at primary station serves as an interface between the host computer and the circuit it serves. The LCU directs the flow of input and output data between the different data communications links and their respective applications program. The LCU performs parallel-to-serial and serial-to-parallel conversion of data and transfers to modem serially. LCU also performs error detection and correction apart from inserting and deleting data link control characters.
When a device has multiple paths to reach a destination, it always selects one path by preferring it over others. This selection process is termed as Routing. Routing is done by special network devices called routers or it can be done by means of software processes.The software based routers have limited functionality and limited scope.In case there are multiple path existing to reach the same destination, router can make decision based on Hop Count, Bandwidth, Metric, Prefix-length or Delay. Routing decision in networks, are mostly taken on the basis of cost between source and destination. Hop count plays major role here. Shortest path is a technique which uses various algorithms to decide a path with minimum number of hops. Common shortest path algorithms are Dijkstra's algorithm, Bellman Ford algorithm or Floyd algorithm. This presentation simplifies Floyd's algorithm with pictures and example.
A digital signal is a sequence of discrete, discontinuous voltage pulses. Each pulse is a signal element. Binary data '0' and '1' are transmitted over digital channel by encoding each data bit into signal elements. Encoding scheme is mapping from data bits to signal elements. Line coding is done to prevent DC wandering and loss of synchronisation on long strings of '0' and '1'. It may give some amount of error detection as in AMT.
Developed by ITU-T, ISDN is a set of protocols that combines digital telephony and data transport services to digitise the telephone network to permit the transmission of audio, video and text over existing telephone line. ISDN is an effort to standardise subscriber services, provide user or network interface and facilitate the inter-networking capabilities of existing voice and data networks. The goal of ISDN is to form a wide area network that provides universal end-to-end connectivity over digital media by integrating separate transmission services into one without adding new links or subscriber links.
Asynchronous Transfer Mode ATM is the cell relay protocol designed by ATM Forum and adopted by the ITU-T. Cell, a small fixed size block of information with asynchronous TDM ensures high speed real time transmission with efficient and cheaper technology. Instead of user addresses, it uses virtual circuit identifier and virtual path identifier, which can be repeated at unrelated locations. This technology ensures connectivity to much more users than normal packet switching networks.
ATM and ISDN-B combination allows high-speed interconnection of world's network.
An artificial Neural Network (ANN) is an efficient approach for solving a variety of tasks using teaching methods and sample data on the principal of training. With proper training, ANN are capable of generalizing and recognizing similarity among different input patterns.The main problem in using ANN is parameter setting, because there is no definite and explicit method to select optimal parameters of the ANN. There are a number pf parameters that must be decided upon like number of layers, number of neurons per layer, number of training iteration, number of samples etc...
Weight enumerators of block codes and the mc williamsMadhumita Tamhane
Best possible error control codes of a certain rate and block length can be adjudged depending on bounds such that no codes can exist beyond the bounds and codes are sure to exist within the bounds. This presentation gives composition structure of Block codes and the probability of decoding error and of decoding failure.Mac William's Identities is relationship between weight distribution of a linear code and weight distribution of its dual code, which hold for any linear code and are based on vector space structure of linear codes and on the fact that dual code of a code is the orthogonal compliment of the code...
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
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Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
2. SETS
Collection of objects.
A set with n objects is -- { s1, s2, …., sn}, where s1, s2, etc are set’s
elements.
A new set formed using any m elements of n elements of a set is
subset of original set.
Finite sets – finite number of elements.
e.g. set of Decimal digits is { 0 1 2 3 4 5 6 7 8 9}
Binary set { 0 1} is subset of decimal set.
Infinite sets – infinite number of elements.
e.g. non-zero positive integers, {1,2,3,….}
Set of real numbers.
Operations between elements within a set can be understood by
higher mathematical structure called “Groups”.
3. GROUPS
A set G on which a binary operation * is defined between elements
within the set is called Group if following conditions are satisfied:-
1. The group is closed under operation *. (Closure property)
1. If a and b are elements of set, then c = a * b also belong to the set.
2. The operation * is associative.
1. a * ( b * c ) = ( a * b ) * c
3. There exists a unique identity element e within the set such that
for any element a,
1. a * e = e * a = a
4. For any element a, there exists unique inverse a’ within the set
such that,
1. a * a’ = a’ * a = e
If a * b = b * a, group is commutative (Abelian Group). It is NOT a
necessary condition for a set to be group.
4. GROUP OPERATIONS
Group operation is normally addition or multiplication.
Gives Additive Groups or Multiplicative Groups.
Example: Binary numbers { 0, 1 } in modulo–2 addition operation.
1. Closure property obeyed.
2. It is associative.
3. Identity element is 0.
4. Each element is its own inverse. ( Addition should result in 0.)
5. It also forms a commutative group.
For coding theory, it is an important group.
5. EXAMPLE
Find whether ordinary addition forms a group over
A) { 0,1,2,…}
B) {0. ±1, ±2, …}
A) No. Addition over this set fails to generate the group.
1. Set is closed. 3 + 4 = 7, a member.
2. Set is associative. (3 + 4) + 7 = 3 + ( 4 + 7) = 14
3. Identity element is 0. Element unaltered when added to 0.
4. But set does not contain inverse of any element. E.g. inverse of 9 is -9.
B) Yes. Addition over this group forms a commutative group.
1. Set is closed. 3 + 4 = 7, a member.
2. Set is associative. (3 + 4) + 7 = 3 + ( 4 + 7) = 14
3. Identity element is 0. Element unaltered when added to 0.
4. Inverse of each element is within the group. E.g. inverse of 9 is -9.
5. Moreover obeys commutative rule.
6. EXAMPLE
Find whether ordinary multiplication forms a group over
A) { 0,1,2,…}
B) {0. ±1, ±2, …}
None form a multiplicative group as group does not contain
multiplicative inverse of the members.
Find whether set of nonzero rational numbers form multiplicative
group?
Yes.
Identity element is 1.
If c = a/b, multiplicative inverse c’ = b/a
Multiplication is closed, associative and commutative too.
Additive group can be created by including 0 element.
7. GROUP OPERATIONS – SUBTRACTION AND DIVISION
Group operation is normally addition or multiplication.
No second group operation definition required within additive
group.
Subtraction can be carried out by replacing element to be
subtracted with its additive inverse.
a – b = a + (-b)
Division can be carried out by multiplicative group by replacing
divisor with its multiplicative inverse.
a / b = a * ( b’)
Associative and commutative properties applies to group elements.
a – b = a + ( - b) = ( - b) + a = - b + a
8. SUB-GROUP
Subset of elements within a group forms a subgroup.
Addition over the set of integers ( a subgroup of rational numbers) form
additive subgroup.
Set of positive integers ( a subset of integers) do not form additive
subgroup. Why?
Set of positive integers do not contain additive inverses.
9. FINITE GROUP
Group having finite number of elements.
Number of members within finite group is called ‘Order” of the
group.
A finite group can be constructed by taking integers modulo-m.
n modulo-m is remainder r obtained after dividing n by m.
Written as n = r modulo-m or n modulo-m = r
5 modulo-4 = 1
13 modulo-5 = 3
10. MODULO-M ADDITION
n = r modulo-m or n modulo-m = r
Modulo-m produces m integers 0,1,2,…,m-1.
If n=m, m modulo-m = 0
If n=a x m, (a x m) modulo-m = 0 ( a is an integer)
If a and b are two integers, using modulo-m addition,--
(a + b ) modulo-m = r
Modulo-m addition set {0,1,2,…,m-1} is commutative group as -
Closed,
Associative,
Commutative ,
Each nonzero integer a has an additive inverse m – a
0 is its own inverse.
11. EXAMPLE-1
Construct the additive group of integers modulo-5 over the set
{0,1,2,3,4}.
Need to find remainders when integers are added pair wise.
(1 + 1) modulo-5 = 2
(1 + 2) modulo-5 = 3
(1 + 3) modulo-5 = 4
(1 + 4) modulo-5 = 0
(2 + 2) modulo-5 = 4
(2 + 3) modulo-5 = 0
(2 + 4) modulo-5 = 1
(3 + 3) modulo-5 = 1
(3 + 4) modulo-5 = 2
(4 + 4) modulo-5 = 3
12. EXAMPLE-1
Adding 0 to any element does not alter element.
Remaining pairs are same as commutative equivalent.
Additive Group of Integers Modulo-5 -
13. EXAMPLE-2
Find inverse of each element in Additive Group of integers modulo-
5.
Additive integer of a nonzero integer a is m – a.
When added to a, it should give 0 after given operation modulo-5.
Additive inverse of 1,2,3, and 4 are 4, 3, 2, and 1.
Additive inverse of 0 is 0.
14. MODULO-M MULTIPLICATION
Gives remainder of the product of two integers.
Modulo-m multiplication generates groups for only prime
values of m.
Group is over the set {1,2,3,…m - 1}.
0 element is excluded as it does not have a multiplicative
inverse.
If m is not a prime number, then modulo-m multiplication
over the set {1,2,3,…m - 1} does not produce a set.
Does not give multiplicative inverse.
Hence, two groups can be constructed over the set
{0,1,2,3,…m - 1},
An additive group over all elements,
A multiplicative group over set’s nonzero elements.
15. EXAMPLE-3
Construct the multiplicative group over the set {1,2,3,4}using
modulo-5 multiplication.
Obtain remainders when integers are multiplied pair wise.
2 X 2 = 4 modulo-5
2 X 3 = 1 modulo-5
2 X 4 = 3 modulo-5
3 X 3 = 4 modulo-5
3 X 4 = 2 modulo-5
4 X 4 = 1 modulo-5
Product with 1 keeps integers unaltered.
Remaining pairs are commutative equivalents.
16. EXAMPLE-3
Construct the multiplicative group over the set {1,2,3,4}using
modulo-5 multiplication.
Multiplicative inverse for a is a’ such that a x a’ = 1 modulo-5.
Inverse of 2 is 3. Find other inverses.
17. PROOF
Prove 1: The identity element in a group G is unique.
Let there exists two identity elements e and e’ in G.
Then e’ = e’ * e = e.
This implies that e and e’ are identical.
Hence there is one and only one identity element.
Prove 2: The inverse of a group element is unique.
Let there exist two inverses a’ and a’’ for group element a.
Then a’ = a’ * e = a’ * (a * a’’) = (a’ * a )* a’’ = e * a’’ = a’’.
This implies that a’ and a’’ are identical.
Hence there is one and only one inverse for a.
18. FIELDS
A set F on which two binary operations, addition and multiplication
are defined between elements within the set is called Field if
following conditions are satisfied:-
1. The Field is a commutative group under addition having
1. Additive identity - zero.
2. Additive inverse.
2. A set of non-zero elements in F is commutative under
multiplication.
1. Having multiplicative identity – unity.
2. Having multiplicative inverse.
3. Multiplication is distributive over addition.
1. a * ( b + c) = a*b + a*c
19. FIELDS
Field consists of at least two elements, AI and MI.
Number of elements in a field is called the ORDER of the field.
Types – Finite field and Infinite field.
Subtraction element b from a is by adding AI (–b) to a.
a – b = a + (-b)
Division of a by b is by multiplying a by MI b-1.
a / b = a * b-1
Hence, within a field, all following are possible-
Addition
Subtraction
Multiplication
Division
20. FIELDS - PROPERTIES
1. For every element a in field, a · 0 = 0 · a = 0
2. For any two nonzero elements, a and b in field, a · b ≠ 0
3. a · b = 0 and a ≠ 0 imply that b = 0
4. For any two elements, a and b in field,
- (a · b) = (- a) · b = a · (- b)
5. For a ≠ 0 , a · b = a · c implies that b = c
21. FIELDS – EXAMPLE –
Set of real numbers
Addition and multiplication form commutative groups.
Two operations satisfy distributive rule.
Gives REAL FIELD.
Set of integers -- { 0, ±1, ±2, ±3, …}
Addition forms a group.
Multiplication FAILS to form a group.
Product of two integers is an integer BUT Division of integer by
another integer is not an integer.
MI not available in this set.
Called a RING.
22. RING
Set of integers -- { 0, ±1, ±2, ±3, …} is a ring.
Addition, subtraction and multiplication are possible but division
not possible.
Structure between Group and Field. Why?
Group-- Either addition and subtraction (AG)or Multiplication and
division (MG).
Field – Addition, subtraction, multiplication and division all are
possible.
23. CONSTRUCTION OF FIELDS
Fields can be created from modulo-m addition and multiplication.
Modulo-2 arithmetic forms AG over set { 0, 1} and MG over non-
zero elements of { 0, 1}.
Generates binary field GF(2)
Finite fields are called Galois fields.
Important for coding theory.
Widely used in digital computers and digital data transmissions.
modulo-2 addition modulo-2 multiplication
+ 0 1 * 1
0 0 1 1 1
1 1 0
24. EXERCISE: CONSTRUCT THE PRIME FIELD UNDER MODULO-7 ARITHMETIC GF(7). FIND
ADDITIVE IDENTITY AND INVERSE AND MULTIPLICATIVE IDENTITY AND INVERSE.
(Prime fields GF(p)– larger fields generated by modulo-m addition and
multiplication over set { 0, 1, …, m-1}, p =m is prime. )
Additive inverse of a– (7 – a)
Additive identity -- 0
Multiplicative inverse of a– a X a* = 1 modulo-m
Multiplicative identity -- 1
25. EXERCISE: EVALUATE ((2 – 3 ) X 3 ) / 4 OVER PRIME FIELD
MODULO-M WHEN
A) M = 5
B) M = 7
A) M = 5
((2 + (-3 ) X 3 ) X (1/ 4)
Additive inverse of 3 is 2 and multiplicative inverse of 4 is 4.
(2 + 2) X 3 X 4
48 modulo-5 = 3 modulo-5
=3
B) M = 7
AI = 4, MI = 2
Ans = 1
27. 1. 3X + 2Y = 2 MODULO - 7
4X + 6Y = 3
Eliminating x by multiplying 1 by 6
18x + 12Y = 12 ≈ 4x + 5y = 5
Solving with 2 we get
Y = 3 – 5 = 3 + 2 = 5
Hence-
3x + 10 = 2
3x + 3 = 2
3x = 2 – 3 = 2 + 4 = 6
x = 6/3 = 6 X 5 = 30 modulo-7 = 2
x = 2, y = 5
2. Ans x = 1, y = 3
28. PROOFS
Prove : If a be nonzero element of a finite field GF(q), then
Prove that a
q-1
= 1
Let b1, b2, …, bq-1 be the q-1 non-zero elements of GF(q).
Hence q-1 elements a.b1, a.b2, …, a.bq-1 are non-zero and
distinct.
(a.b1) . (a.b2) …(a.bq-1) = b1. b2. … bq-1 (Try multiplying a=2 with all
nonzero elements of GF(7) using mod-7)
a
q-1
(b1 . b2 …bq-1) = b1. b2. … bq-1
a
q-1
= 1
29. PROOFS
Prove : If a be nonzero element of a finite field GF(q), and n
be the order of a then,
Prove that n divides q-1
Let n does not divide q-1.
Dividing q-1 by n , we obtain q-1 = kn +r where 0 < r < n
Then, a
q-1
= a
kn + r
= a
kn
. a
r
= (a
n
)
k
. a
r
Since a
q-1
=1 and a
n
=1 as order of a is n—
a
r
=1
This is impossible as 0 < r < n and n is smallest integer such
that a
n
=1.
Hence assumption is wrong.
n must divide q-1.
30. PRIME FIELDS
For any prime p, there exists a finite field of p elements GF(p).
For any positive integer m, it is possible to extend the prime
field GF(p) to a field of pm
elements called extension field
GF(pm
).
Order of any prime field is the power of the prime.
31. BINARY FIELD ARITHMETIC
Codes can be constructed with symbols from any Galois field
GF(p), where p is either prime or a power of p.
Codes most widely used in data communication use p = 2.
GF(2) or its extension GF(2m)
GF(2) uses binary arithmetic – modulo-2 addition and
multiplication.
1 + 1 = 0, 1 = -1 → Subtraction is same as addition.
Binary arithmetic can be used to solve set of equation using
Cramer’s rule. As
X + Y = 1
X + Z = 0
X + Y + Z = 1
33. POLYNOMIAL OVER GF(2) (ONE VARIABLE)
Degree of polynomial is largest power of X with nonzero
coefficient.
Polynomial over GF(2) takes its coefficients from GF(2).
Total of 2n
polynomials over GF(2) with degree n.
If n = 1, → X and 1 + X
If n = 2 → X2
, 1 + X2
, X + X2
, 1+ X + X2
Polynomial over GF(2) can be added, subtracted, multiplied and
divided. Using modulo-2 addition and multiplication.
35. POLYNOMIAL OVER GF(2) (ONE VARIABLE)
Polynomials can be multiplied over GF(2).
When f(X) is divided by g(X), we get, using Euclid’s division algorithm,
Degree of r(X) is less than that of g(X),
36. POLYNOMIAL OVER GF(2) (ONE VARIABLE)
Polynomial over GF(2) follow following conditions:
37. POLYNOMIAL OVER GF(2) (ONE VARIABLE)
f(X) = 1 + X + X
4
+ X
5
+ X
6
g(X) = 1 + X + X
3
Find f(X) · g (X) = ?
1 + X
2
+ X
3
+ X
8
+ X
9
.
Find f(X) / g (X) = ?
1 + X + X
2
is remainder and quotient is X
2
+ X
3
.
38. ROOTS OF POLYNOMIAL OVER GF(2)
If a is root of polynomial f(X), f(X) is divisible by (X – a).
If f(X) = 1 + X
2
+ X
3
+ X
4
, and X = 1, f(1) = 0
Hence 1 is root and f(X) is divisible by (X + 1).
If a Polynomial over GF(2) has even number of terms, it is divisible by
(X + 1). Why? 1 is root.
Polynomial over GF(2) of degree m is called Irreducible over GF(2), if
it is not divisible by any Polynomial over GF(2) of degree less than m
but greater than zero. (Divisible by only 1 and self.)
If degree m = 2, then
X
2
, 1 + X
2
, and X + X
2
are divisible by X or 1 + X, → roots 0 and 1.
but 1+ X + X
2
is not divisible by any polynomial of degree 1
1+ X + X
2
is irreducible polynomial of degree 2.
1+ X + X
3
is irreducible polynomial of degree 3.
1+ X + X
4
is irreducible polynomial of degree 4.
40. CONSTRUCTION OF GALOIS FIELD GF(23)
As seen before, polynomial 1+ X + X3
has no binary (0,1) roots.
Let , not belonging to binary field GF(2) but lying within finite
field GF(23
) is one of the roots of above polynomial.
Hence 3
+ + 1 = 0
Elements of GF(23
) = ?
0 and 1 form additive and multiplicative identity elements.
+0 = · 1 =
Additive inverse of is itself.
+ = 1 + 1 = (1 + 1) = 0 · = 0
Subtraction and addition of are equivalent
From above = -
Multiplicative inverse of = -1
= 1/ .
The other elements of field GF(23
) can be generated from .
42. FIELD ELEMENTS OF GALOIS FIELD GF(2M)
We can see higher powers as
7
= 1 ( 7 = 0 modulo-7)
8
= ( 8 = 1 modulo-7)
9
= 2
(9 = 2 modulo-7)
And so on…
12
= 5
(12 = 5 modulo-7)
17
= 3
(17 = 3 modulo-7)
GF(23
) has 8 basic elements- 0, 1, , 2
, 3
, 4
, 5
, and 6
Degree of polynomial = m = 3
Field’s characteristics = no, of elements in base field = p = 2
Order of the field = q = p
m
.
Coefficients of polynomials belong to base field while roots belong to
extension field.
46. CONSTRUCTION OF GALOIS FIELD GF(24)
As seen before, polynomial 1+ X + X
4
has no binary (0,1) roots.
Let , not belonging to binary field GF(2), but lying within finite
field GF(24
) is one of the roots of above polynomial.
Hence 4
+ + 1 = 0
The field GF(24
) can be generated from .
0 and 1 form additive and multiplicative identity elements.
+1 = · 1 =
Additive inverse of is itself.
+ = 1 + 1 = (1 + 1) = 0 · = 0
Subtraction and addition of are equivalent
From above = -
Multiplicative inverse of = -1
= 1/ .
49. PRIMITIVE FIELD ELEMENTS
Field element that can generate all the nonzero elements of
the field are primitive elements.
is primitive in GF(23
), GF(24
) and GF(25
).
All elements except 0 and 1 are primitive for GF(23
). e.g…
Show that
2
is primitive in GF(2
3
).
Let = 2
,
2
=( 2
)2
= 4
3
=( 2
)3
= 6
4
=( 2
)4
= 8
=
5
=( 2
)5
= 10
= 3
6
=( 2
)6
= 12
= 5
7
=( 2
)7
= 14
= 1
8
=( 2
)8
= 16
= 2
And repeats.
50. PRIMITIVE FIELD ELEMENTS
Show that
5
is primitive in GF(2
3
).
Show that 2
is primitive in GF(24
).
Show that
3
is NOT primitive in GF(2
4
).
Show that 5
is NOT primitive in GF(24
).
51. ORDER OF THE ELEMENTS
Smallest positive integer n for which
n
= 1
Determines if is primitive or not.
(Not same as order of the field, which is number of elements
within field.)
In GF(23
), all field elements have same order 7. Show.
In GF(24
), all field elements do not have same order . Show.
Order of an element in GF(2m
) divides 2m
- 1
For GF(24
) determine the order of 12
and 7
. Find if they are
primitive or not. What field elements they generate?
Smallest power of
12
to give unity is 5. Not primitive as 15
elements are required. Elements – 12 9 6 3
1
Smallest power of
7
to give unity is 15. Primitive as 15
elements are required. Elements – all
52. IRREDUCIBLE AND PRIMITIVE POLYNOMIAL
Polynomials, divisible by only 1 and self are called irreducible
polynomials.
Irreducible polynomial having a primitive field element as a root
is called a primitive polynomial.
An irreducible polynomial p(X) of degree m is said to be primitive
if the smallest positive integer n for which p(X) divides X
n
+ 1 is
n=2m
– 1. (Not any n <2
m
– 1)
The irreducible polynomial p(X) = X4
+ X + 1, divides (X15
+ 1)
(n=15), but does not divide any Xn
+ 1 for 1<n<15, hence p(X) is
primitive polynomial.
The polynomial p(X) = X4
+ X3
+ X2
+ X + 1 is irreducible but not
primitive as it divides X5
+ 1 also.
53. CONJUGATE OF FIELD ELEMENT OVER GF(2
M
)
In ordinary algebra, a polynomial may have complex conjugate
roots occurring in pair.
Similarly roots of polynomial with coefficients from GF(2) also
occur In groups or sets of conjugates.
X4
+ X3
+ 1 has no roots from GF(2), but has 4 roots from GF(24
).
By substitution, roots are -
7
,
11
,
13
and
14
. Verify
Then (X + 7
)(X + 11
)(X + 13
)(X + 14
) = X4
+ X3
+ 1
If one root is known, its other conjugate roots can be found as—
54. CONJUGATE OF FIELD ELEMENT OVER GF(2
M
)
If one root is known, its other conjugate roots can be found as—
Theorem- Let f(X) be a polynomial with coefficients from GF(2).
Let β an element in an extension field of GF(2). If β is a root of
f(X), then for any l ≥ 0, β2
l
is also a root of f(X).
Proof: We have [f(X)] 2
l
= f(X 2
l
)
β is Root --- [f(β)] 2
l
= f(β2
l
)
Since f(β)= 0, f(β2
l
) = 0
Hence β2
l
is also a root of f(X).
The element β2
l
is called a conjugate of β.
Hence if β from GF(2m
) is a root of f(X) over GF(2), then all
conjugates of β ( which are elements of GF(2
m
),are also roots of
f(X).
55. CONJUGATE OF FIELD ELEMENT OVER GF(2
M
)
Example – Let f(X) = 1+ X3
+ X4
+ X5
+ X6
has 4
from GF(24
), as a
root. Verify. Also find its conjugate roots.
f( 4
) = 0.
The conjugates of 4
are –
( 4
)2
= 8
,
( 4
) 22
= 16
=
( 4
) 23
= 32
= 2
. Higher powers repeat roots.
Check.
Find other two roots. Are they conjugates?
5 10
. They are conjugates.
56. THEOREM
Theorem2.8- The 2m
- 1 non zero elements of GF(2m
) are all the
roots of (X(2m– 1)
+ 1).
Proved earlier that -
If β is nonzero element in the field GF(2m
),---- β2m-1 =1
Example – For GF(24
) , β15 = 1, for β = 1, 2 3 4 5 6
…… 14
Adding 1 on both sides,
β2m-1 +1 = 0
Hence β is the root of X2m-1 +1 = 0
Hence all 2
m
-1 nonzero element are roots of X2m-1 +1 = 0
Prove Corollary 2.8.1– The elements of GF(2m
) form all the roots
of X2m
+X
X2m
+X = X (X2m-1 +1)
Hence element 0 is also a root.
57. MINIMAL POLYNOMIAL
Let any element β in GF(2m
) is a root of (X(2m– 1)
+ 1) over GF(2).
β may also be a root of a polynomial over GF(2)with degree < 2m
.
The binary polynomial of smallest degree, of which β is a root, is
called minimal polynomial of β.
Theorem 2.9: Minimal polynomial φ(x) of a field element β is
irreducible.
Suppose φ(x) is NOT irreducible. Then φ(x) = φ1(x) φ2(x)
Both φ1(x) and φ2(x) have degrees > 0 and < degree of φ(x)
As φ(β) = φ1(β) φ2(β) = 0
Either φ1(β) = 0 or φ2(β) = 0
This contradicts the hypothesis that φ(x) is a polynomial of
smallest degree, such that φ(β) = 0.
Therefore φ(x) must be irreducible.
58. MINIMAL POLYNOMIAL
Theorem 2.10: Let f(x) be a polynomial over GF(2). Let φ(x) be the
minimal polynomial of a field element β. If β is the root of f(x),
then f(x) is divisible by φ(x).
Dividing f(x) by φ(x),
f(x) = a(x) φ(x) + r(x)
Degree of r(x) < degree of φ(x).
If x= β ,then f(β) = a(β) φ(β) + r(β)
As f(β) = φ(β) = 0
r(β) = 0
If r(x) ≠ 0, r(x) would have degree < φ(x), which has β as root.
This contradicts fact that φ(x) is a minimal polynomial of β.
Hence r(x) must be zero and φ(x) divides f(x).
59. MINIMAL POLYNOMIAL
Theorem 2.11: Minimal polynomial φ(x) of a field element β in
GF(2m
) divides (X2m
+ X).
All roots of φ(x) are from GF(2m
).
Theorem 2.8 corollary 2.8.1 says field elements of GF(2m
) are
roots (X2m
+ X).
β is root of φ(x) as well as (X2m
+ X).
As φ(x) is minimal polynomial, from theorem 2.10, φ(x) divides
(X2m
+ X).
60. MINIMAL POLYNOMIAL
Theorem 2.12: Let f(x) be an irreducible polynomial over GF(2).
Let β be an element in GF(2m
). Let φ(x) be the minimal
polynomial of element β. If f(β) = 0 then φ(x) = f(x).
From 2.10, φ(x) divides f(x).
Since φ(x) ≠ 0, and f(x) is irreducible, φ(x) = f(x).
61. MINIMAL POLYNOMIAL
Theorem 2.13: Let β be an element in GF(2m
) and Let e be the
smallest non-negative integer such that β2e
= β. Then
f(X) = ∏i=0
e-1
(X + β2i
) is an irreducible polynomial over GF(2).
62. MINIMAL POLYNOMIAL
For 0 ≤ i ≤ e, fi = fi
2
.
It is possible only when , fi = 0 or 1.
Hence f(x) Has coefficients from GF(2).
Suppose f(X) is NOT irreducible over GF(2) and f(X) = a(X) b(X).
Since f(β) = 0, either a(β) = 0 or b(β) = 0.
Hence
63. MINIMAL POLYNOMIAL
If a(β) = 0, a(X) has , β, β2, …β2e-1
as roots, so
a(X) has degree e and
a(X) = f(X).
Similarly if b(β) = 0, b(X) has , β, β2, …β2e-1
as roots, so
b(X) has degree e and
b(X) = f(X).
This contradicts itself.
Hence f(X) must be irreducible.
64. MINIMAL POLYNOMIAL
Theorem 2.14 : Let ¢(X) be the minimal polynomial of an element
β in GF(2m
). Let e be the smallest integer such that β2e
= β. Then
¢(X) = ∏i=0
e-1
(X + β2i
)
From 2.13, f(X) is irreducible.
¢(X) is irreducible minimal polynomial.
Hence ¢(X) = ∏i=0
e-1
(X + β2i
)
65. EXAMPLE
For Galois Field GF(24
), β = 3
. Find conjugates of β and the
minimal polynomial of β = 3
.
Conjugates are – 6
, 12
, 24
= 9
.
Minimal polynomial ¢(X) =(X + 3
) (X + 6
)(X + 12
)(X + 9
).
= 1+ X + X2
+ X3
+ X4
For Galois Field GF(23
), find minimal polynomial of 3
.
66. COMPUTATION USING GALOIS FIELD GF(2M
) ARITHMETIC.
Given linear equations over GF(24
) as
X + 7
Y = 2
12
X + 8
Y = 4
Multiply eq2 with 3
and add the two.
X + 7
Y = 2
X + 11
Y = 7
( 7
+ 11
) Y= 2
+ 7
8
Y= 12
Y = 4
X = 9
Alternately use Cramer’s rule.
67. COMPUTATION USING GALOIS FIELD GF(2M
) ARITHMETIC.
Find roots of X2
+ 12
X + 9
= 0 for GF(24
)
Let β1 ,β2 are required roots.
X2
+ X(β1+β2 ) + β1 β2 = 0
β1+β2 = 12
β1 β2 = 9
Roots that satisfy = 2
, 7
. Check by finding minimal polynomial.
Expand (X + 4
)2
in GF(23
).
=X2
+ X 4
+ X 4
+ 8
= X2
+ 8
= X2
+ .
Expand (X + 3
)5
(X + 10
) in GF(24
).
(X + 3
)4
(X + 3
) (X + 10
)
(X4
+ 12
) (X2
+ 12
X + 13
)
X6
+ 12
X5
+ 13
X4
+ 12
X2
+ 9
X + 10
)
68. COMPUTATION USING GALOIS FIELD GF(2M
) ARITHMETIC.
Find inverse of following in GF(24
).
2 13
0 10
7 3
1
14
0 12
5 13
1
14 9
Find inverse of following in GF(24
).
3 5
1
8 12
7 10
69. COMPUTATION USING GALOIS FIELD GF(2M
) ARITHMETIC.
Find solution of following in GF(24
) using Cramer’s rule.
3
X + Y +Z = 5
2
X + 6
Y +Z = 6
14
X + 7
Y + 7
Z = 1
X =
Y = 4
Z = 4
70. RELATION WITH CYCLIC CODES
Code word c(x) for an (n,k) cyclic code is c(x) = f(x) g(x)
Nonsystematic CRC – f(x) is data polynomial d(x)
Systematic CRC – f(x) is quotient q(x) after dividing d(x) xn-k
by
g(x)
Root of g(x) is also root of c(x).
Let for (7,4) CRC, g(x) = 1+ X + X3
and c(x) = 1+ X + X2
+ X5
Roots , 2
and 4
of g(x) belonging to GF(23
) are roots of c(x).
Polynomials like 1+ X + X3
generate CRC as well as construct
finite field.
Generator polynomial can be obtained by finding minimal
polynomial using all roots and finding LCM to eliminate
common multiples.
Find generator polynomial using , 2
and 4
in GF(23
) .
71. VECTOR SPACES
Code word belonging to (n.k) block code can be interpreted as
vectors within k-dimensional vector space.
Vector space is defined as collection of objects, called vectors,
together with operations of vector addition and scalar
multiplication satisfying following conditions :-
1. Set of vectors forms additive cumulative group satisfying
closure, associative and commutative property and has
additive inverse and additive identity vector 0.
2. Multiplication of vectors with scalar is defined. Scalar product
is a vector with distributive and associative property and has
multiplicative identity 1.
72. ORDERED SEQUENCES – N-TUPLE
Vector v = (v1, v2, ….vn)
Where vi for i = 1,2,…n, are scalars from a field and are
components of v.
Vector addition is component wise.
If u = (u1, u2, ….un)
w = u +v is
w = (u1 + v1, u2 + v2, … un + vn)
= (w1, w2, ….wn)
Scalar product is also component wise.
au = (au1, au2, …aun)
73. EXAMPLE
Given vectors u = (2,7,1) and v = (10,-3,8),
Find
1. u + v
2. 4u
3. 2u – 7v
Answer-
1. (12, 4, 9)
2. (8, 28, 4)
3. (-66, 35, -54)
74. LINEARLY DEPENDENT VECTORS
A set of vectors v1, v2, ….vm are said to be linearly
dependent if there exists m scalars a1, a2, …am, not all
of them zero, such that— a1v1 + a2v2 + ….amvm = 0
Otherwise vectors are said to be linearly independent.
For linearly independent vectors, only scalars that
satisfy above equation are zeros.
Find if following ser of vectors are linearly dependent.
1. u1 = (3, -8, 5), u2 = (-2, 2, 14) and u3 = (-1, 6, -19)
2. v1 = (7, 3, -3), v2 = (16, 2, -1) and v3 = (6, -12, 15)
3. w1 = (-2, 0, 0), w2 = (0, 7, 0) and w3 = (0, 0, 9)
75. ANSWER
1. If a1 = 1, a2 = 1 and a3 = 1
• a1u1 + a2u2 + ….amum = 0
• Linearly dependent
2. If a1 = 1, a2 = 1 and a3 = 1
• a1v1 + a2v2 + ….amvm ≠ 0
• But if a1 = 2, a2 = -1 and a3 = 1/3
• a1u1 + a2u2 + ….amum = 0
• Linearly dependent
3. No combination of scalars satisfy condition.
1. Linearly independent
76. IMPORTANCE OF LINEAR INDEPENDENCE OF VECTORS
Ability to form linear combination and linear independence
simplify construction of vector spaces.
Within a vector space, there exists a set of linearly independent
vectors, from which all other vectors can be generated.
This set is called Basis and these vectors, Basis Vectors.
Each vector in a vector space is unique combination of basis
vectors.
Basis vectors are said to span the vector space.
Number of vectors in a Basis is called Dimension of vector
space.
In m-dimensional vector space, m linearly independent vectors
span the space.
77. EXAMPLE
Three dimensional space over the real field.
Basis Vectors are—
i = ( 1 0 0)
j = ( 0 1 0)
k= ( 0 0 1)
Also called Standard Basis because --
All other vectors within space can be expressed using them as
–
v = vxi + vyj + vzk
v = vx ( 1 0 0) + vy ( 0 1 0) + vz ( 0 0 1)
v = (vx vy vz)
What is Standard Basis for V4?
78. BINARY N-TUPLE
n bit vectors made with 0s and 1s.
A vector space Vn is formed by 2n n-tuples using modulo-2
addition.
Example - vector space V4 is formed by 16 binary 4-tuples.
From (0 0 0 0) to (1 1 1 1)
(0 0 0 0) is identity element and each vector is its own inverse.
Scalars are 0 and 1.
Vector addition and scalar multiplication obey required
associative, distributive and commutative laws.
79. VECTOR SUBSPACE
Subset of vectors, existing within a vector space, having all the
characteristics of vector space under vector addition and scalar
multiplication.
example – following four vectors under V4.
(0 0 0 0), (0 1 1 1), (1 0 1 0), (1 1 0 1)
The subset follows closure, commutative, associative and
distributive property under vector addition and scalar
multiplication .
Has additive identity and inverse.
Hence forms subspace of V4.
One more subspace of V4:-
(0 0 0 0), (0 1 1 0), (0 0 1 0), (0 1 0 0)
Standard Basis of Vn can not be a basis for a subspace. Why?
80. CREATION OF VECTOR SPACE FROM MATRICES
Given n X m matrix as-
a1,1
a1,2
….. v1
a2,1
a2,2
…. v2
an,1
an,2
… vn
where ai,j
= 0 or 1.
Let v1
, v2
, …vn
are row vectors ( or row spaces) of matrix, then
set of all linear combinations of row vectors forms vector space.
Interchanging rows and columns also give same vector space.
Example :- Let matrix A =
1 0 0 1 1
0 1 0 1 0
0 0 1 0 1
Find row spaces and vectors in vector spaces.
Row spaces v1
= 1 0 0 1 1 , v2
= 0 1 0 1 0 v3
= 0 0 1 0 1
82. CREATION OF VECTOR SPACE FROM MATRICES
Exercise:- Form matrix B from A by adding row 3 to row 1 and
then interchanging rows 2 and 3.
Find row spaces and vectors in vector spaces. Show that they
are same as for A.
83. LINEARITY OF CODES BASED ON VECTOR SPACES
For (n,k) code, two sets of words are defined.
SET I : 2n
n-bit words,
form vector space Vn,
its elements form commutative group
Vector addition and scalar multiplication satisfy associative,
commutative and distributive conditions.
SET II: smaller set Ck formed by 2k
code words.
LINEARITY requirement:-
Code words Ck forms subspace over vector space Vn.
Subspace over vector space is also a vector space.
There exist at least one set of k linearly independent basis
vectors which span the vector space such that vectors in V
can be formed by linear combinations of basis vectors.
84. LINEARITY OF CODES BASED ON VECTOR SPACES
For a linear code, rows of generator matrix G form a set of basis
vectors for the code.
Taking all combinations of rows of G code words can be
generated.
85. LINEARITY OF CODES BASED ON VECTOR SPACES
Example: The (6,3) code has generator matrix
1 0 0 0 1 1
0 1 0 1 0 1 = G
0 0 1 1 1 0
Show that the codewords can be generated by taking their
linear combinations of the rows.
Row 1 = (1 0 0 0 1 1)
Row 2 = (0 1 0 1 0 1)
Row 3 = (0 0 1 1 1 0)
Row 1+ Row 2= (1 1 0 1 1 0)
Row 1+ Row 3= (1 0 1 1 0 1)
Row 2+ Row 3= (0 1 1 0 1 1)
Row 1 +Row 2 +Row 3 = (1 1 1 0 0 0 )
Along with 0 0 0 0 0 0) , 8 codewords.
86. CODES BASED ON VECTOR SPACES
Example: The (6,3) code has generator matrix
g1
=1 0 0 0 1 1
g2
= 0 1 0 1 0 1 = G
g3
= 0 0 1 1 1 0
Show that row vectors of G are linearly independent
Check all 8 combinations of g1
,g2
,g3
to find for which
condition, following is satisfied.
a1
g1
+ a2
g2
+ a3
g3
=0
87. CODES BASED ON VECTOR SPACES
Codeword for information word d = d1d2d3d4 is expressed as
c = dG = d1
g1
+ d2
g2
+ d3
g3
+ d4
g4
Equivalent to adding those rows of G for which dk
= 1
Same example – d = 1101
Code can be obtained by multiplying matrix d with G
Or by adding rows 1,2 and 4.
Check.
Example:
1 0 0 0 1 0 1
0 1 0 0 1 1 1 = G
0 0 1 0 1 1 0
0 0 0 1 0 1 1
Find code words by adding rows and check.