This document provides information and formulas related to error control coding and binary cyclic codes. It begins with defining the code vector, message vector, parity check matrix, and generator matrix for error control coding. It then provides the parameters and capabilities for single error correcting Hamming codes. Finally, it presents formulas for non-systematic and systematic cyclic codes, including defining the remainder polynomial, parity check polynomial, and code vector formulations.

1. INFORMATION THEORY AND
CODING
5th SEM E&C
JAYANTHDWIJESH H P M.tech (DECS)
Assistant Professor – Dept of E&C
B.G.S INSTITUTE OF TECHNOLOGY (B.G.S.I.T)
B.G Nagara, Nagamangala Tq, Mandya District- 571448

2. FORMULAS FOR REFERENCE
MODULE – 4 (error control coding and binary cyclic codes)
Chapter -1 error control coding
 The code vector is given by
[C]= [D] [G]
 The Message vector is given by
𝟐 𝒌
 The Parity check matrix is given by
[H]= [ 𝑷 𝑻
: 𝑰 𝒏−𝒌 ] or [ 𝑰 𝒏−𝒌 : 𝑷 𝑻
]
 The generator matrix is given by
[G]= [𝑰 𝑲: P] or [𝑷 ∶ 𝑰 𝑲]
 The syndrome is given by
S=R𝑯 𝑻
or r𝑯 𝑻
 The corrected code vector is given by
C=R+E
The single error correcting (n, k) Hamming code has the following parameters
 Code length: n 𝟐 𝒏−𝒌−𝟏
 Number of message bits : k n-𝒍𝒐𝒈 𝟐 (n+1)
 Number of parity check bits: (n-k)
 Error correcting capability: t =
𝒅 𝒎𝒊𝒏 − 𝟏
𝟐
 Error correcting capability: 𝒅 𝒎𝒊𝒏 − 𝟏
 𝑯 𝑻
is given by
𝑯 𝑻
= [
𝑷
𝑰 𝒏−𝒌
] or [
𝒅 𝒌 𝑿 (𝒏−𝒌)
𝑰 𝒏−𝒌
]
HAMMING BOUND (perfect code or not)
 𝟐 𝒏−𝒌
 𝐂𝐢
𝐧𝒕
𝒊=𝟎

3. Chapter-2 binary cyclic codes
 The Non systematic cyclic code vector is given by
V(X) =D(X) g(X)
 The systematic cyclic :
The remainder polynomial R(X) is given by
R(X) =
𝑫 𝑿 𝑿 𝒏−𝒌
𝒈(𝑿)
The code vector is given by
[V] = [R][D]
 The parity check polynomial is given by
h (x) =𝒙 𝟕
+1 / g(x)
 The code vector is given by
[V] = [R][D]