Partial generation of 2n length walsh codes using n-bit gray and inverse gray c

International Journal of Electronics and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 4, July-August (2013), © IAEME
232
PARTIAL GENERATION OF 2n
-LENGTH WALSH CODES USING n-BIT
GRAY AND INVERSE GRAY CODES
K.USHA1
, Dr. K. JAYA SANKAR2
1
(Dept. of ECE, MVSR Engg. College, Hyderabad, India)
2
(Dept. of ECE, Vasavi College of Engg., Hyderabad, India)
ABSTRACT
This paper presents a technique for the construction of 2n Walsh Code words of length 2n
.
Walsh codes are linear phase and zero mean with unique number of zero crossings for each sequence
within the set. Walsh Codes are fixed length orthogonal codes possessing high auto correlation and
low cross correlation properties. An n-bit Cyclic Gray Code is a circular list of all 2n
bit strings such
that successive code words differ in only one bit position. An ’n’ bit Inverse Gray Code , is defined
exactly opposite to Gray code, it is a circular list of all 2n
bit strings of length ‘n’ each, such that
successive code words differ in (n-1) bit positions. Algorithms discussed in this paper for the
generation of Gray and Inverse Gray codes result in n! sequence orderings. The proposed technique
allows us to construct n! sequence orderings of 2n Walsh Code words (of any length) since they are
constructed from n-bit Gray and Inverse Gray Codes.
Keywords: Gray Code, Inverse Gray Code, Walsh Code.
1. INTRODUCTION
Walsh Code is a group of spreading codes having good autocorrelation properties and poor
cross-correlation properties [1]. Walsh Codes are commonly used as Pseudo random noise (PN)
sequences in Direct Sequence Spread Spectrum (DS-SS) communications. Walsh codes are the
backbone of CDMA systems and are used to develop the individual channels in CDMA. IS-95 uses
64-length Walsh Code set and these allow the creation of 64 channels from the base station.
Excluding the pilot and sync channels a base station can talk to a maximum of 54 mobiles at the
same time. Among the former techniques for the generation of Walsh Codes the popular method is
based on the simple iterative scheme from the Hadamard matrices. This paper presents a technique
for the construction of Walsh codes using n-bit Gray and Inverse Gray codes. An n-bit Gray code is a
list of all 2n
bit strings such that successive code words differ in only one bit position [2, 3]. If the
first and last code words also differ in one bit position then the resultant code is called cyclic. Gray
codes have the adjacency property which makes the hamming distance between adjacent code words
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233
always equal to 1. Inverse Gray codes, as defined in [4], on the contrary, exhibit maximum possible
hamming distance (n-1) between the two successive code words. A very commonly used method of
generating n-bit Gray code from binary is by performing bit-wise XOR operation of two successive
bits. The encoding method in [4] to generate Inverse Gray code is similar to the Gray code
generation technique mentioned above. In this an n-bit Inverse Gray code is derived from the binary
representation [b1, b2, b3…. bn] of an integer ‘h’ from a set of all the integers {0,1,2,…..(2n
-1)}. If ‘n’
is even b0 = 0 else b0 = b1.Inverse Gray code is obtained by performing the bit-wise XOR or XNOR
of bi-1 & bi depending on whether ‘h’ is even or odd respectively. Adopting the encoding technique
described above for odd values of ‘n’ results in a code with the code words in the bottom half of the
list similar to the code words in the top half of the list. So, the Inverse Gray code for ‘n’ odd is
obtained by complementing the Most Significant Bit (MSB) of bottom half. Inverse Gray codes
combined with Robust Symmetrical Number System (RSNS) find applications in the area of error
detection and correction [4].
Inverse Gray Code generation algorithm discussed in this paper is obtained by suitably
modifying the Binary Cyclic Gray Code generation technique in [3]. Inverse Gray code generation
algorithm proposed by the authors is reported earlier [5]. A total of n! sequence orderings of Inverse
Gray codes can be generated for any integer ‘n’. For even and odd values of ‘n’ there is a small
difference in the generation procedure. Walsh code generation using 4-bit Gray and Inverse Gray
codes is reported by the authors earlier [6].
The paper is organized in the following manner: Section 2 briefly discusses the Binary Cyclic
Gray Code generation algorithm in [3]. In Section –3 the Inverse Gray Code generation algorithm is
discussed. The procedure for the construction of Walsh Code sets from n-bit Gray and Inverse Gray
codes is explained in Section-4. And finally, Section-5 concludes the paper with the merits and
demerits.
2. ALGORITHM TO GENERATE BINARY CYCLIC GRAY CODES n - BIT CYCLIC
GRAY CODE (radix r=2), M = 2n
Let an n-bit Cyclic Gray code be needed. Let (P1, P2, P3 ,……….Pn) be a permutation of
(1,2,3,….n). The M = 2n
integers (0, 1, 2, ……, (2n
-1))can be arranged in the following indexed
indicial sets.
Q0 = 20
{1, 3, 5…….}
Q1 = 21
{1, 3, 5…….}
:
:
Qn-1 = 2n-1
Qn = 2n
Then, for any integer value of ‘n’, starting with the row of all zeros as a zeroeth row, the ith
row is obtained from the (i-1)th
row by replacing the pj
th
bit by its successor, if it is in Qj-1.
Let us consider the construction of a 3-bit binary Gray code. All the integers, i.e.,
{0,1,2,3,…..(23
–1)} are arranged in the form of indicial sets as shown below:
Q0 = 20
{1, 3, 5,7} = 1,3,5,7
Q1 = 21
{1, 3} = 2,6
Q2 = 22
{1} = 4

234
As stated earlier, let (P1, P2, P3 …… Pj ……Pn) be a permutation of (1,2,3,….. j,….n). Since
we are considering a 3-bit case, consider the permutation {2,3,1}. Hence, P1 = 2; P2 = 3; P3 = 1. The
first code word is (0 0 0) which is the zeroeth row of the code. To obtain 1st
row, we have to change
Pj
th
bit if ‘1’ is in Qj-1
.
Here, 1 is in Q0. Therefore, P1 bit is to be changed and P1=2, hence the code is
(0 1 0). Similarly, since ‘2’ is in Q1, P2 bit (i.e. 3rd
bit) is to be changed, hence the code is (1 1 0).
The resulting code obtained by continuing this procedure is tabulated in Table I.
Table I: 3-bit Cyclic Gray Code for permutation {2, 3, 1}
Sl
No.
ith
row
Pj
Bit to be
changed
3-bit
Binary
Gray Code
3 2 1
1 0 - - 0 0 0
2 1 P1 2 0 1 0
3 2 P2 3 1 1 0
4 3 P1 2 1 0 0
5 4 P3 1 1 0 1
6 5 P1 2 1 1 1
7 6 P2 3 0 1 1
8 7 P1 2 0 0 1
A total of n! Gray codes can be generated using the above technique for any integer value of
‘n’ and all these Gray codes are cyclic.
Table II: All the 3! = 6, 3-bit Cyclic Gray Codes generated using the algorithm in decimal notation
Permutation Cyclic Gray code
1 1 2 3 0,1,3,2,6,7,5,4
2 1 3 2 0,1,5,4,6,7,3,2
3 2 1 3 0,2,3,1,5,7,6,4
4 2 3 1 0,2,6,4,5,7,3,1
5 3 1 2 0,4,5,1,3,7,6,2
6 3 2 1 0,4,6,2,3,7,5,1
3. GENERATION OF BINARY INVERSE GRAY CODES n - BIT INVERSE GRAY CODE
(radix r=2), M = 2n
Let an n-bit Inverse Gray code be needed. Let (P1, P2, P3 ,……….Pn) be a permutation of
(1,2,3,….n). The 2n
integers (0, 1, 2, ……, (2n
-1))can be arranged in the following indexed indicial
sets.
Q0 = 20
{1, 3, 5…….}
Q1 = 21
{1, 3, 5…….}
:
:
Qn-1 = 2n-1
Qn = 2n

235
Then, for ‘n’ even, starting with the row of all zeros as a zeroeth row, the ith
row is obtained from the
(i-1)th
row by replacing all other bits except the pj
th
bit by its successor, if it is in Qj-1. And for ‘n’
odd, the above procedure is used to obtain all the rows except M/2 th row. For M/2 th row, all the
bits have to changed irrespective of where it falls within the indicial sets.
3.1 Inverse Gray code for ‘n’ even:
Let us consider the construction of a 4-bit Inverse Gray code. All the integers, i.e., {0, 1, 2,
3,…..(24
–1)} are arranged in the form of indicial sets as given below:
Q0 =20
{1, 3, 5,7,9,11,13,15}= 1,3,5,7,9,11,13,15
Q1 = 21
{1, 3,5,7} = 2,6,10,14
Q2 = 22
{1,3} = 4,12
Q3 = 23
{1} = 8
As stated earlier, let (P1, P2, P3 …… Pj ……Pn) be a permutation of (1,2,3,….. j,….n). Since
we are considering a 4-bit case, consider the permutation {1,2,3,4}. Hence, P1 = 1; P2 = 2; P3 = 3; P4
= 4. The first code word is (0 0 0 0) which is the zeroeth row of the code. To obtain 1st
row, we
have to change all other bits except Pj
th
.
Here, 1 is in Q0. Therefore, retaining P1
bit as it is all other bits are to be changed, since P1 = 1 1st
is retained and 2nd
, 3rd
and 4th
bits are
changed. Hence the resultant codeword of the 1st
row is (1 1 1 0). Similarly, to obtain 2nd
row since
‘2’ is in Q1, P2 bit is to be unchanged, hence the code is (0 0 1 1). Table III shows the resulting code
obtained by continuing this procedure.
Table III: 4-bit Inverse Gray Code for permutation {1, 2, 3, 4}
Element
No
i th
row
Pj
Bits to be
Changed
4-bit Inverse Gray
code
4 3 2 1
1 0 - - 0 0 0 0
2 1 P1 2,3,4 1 1 1 0
3 2 P2 1,3,4 0 0 1 1
4 3 P1 2,3,4 1 1 0 1
5 4 P3 1,2,4 0 1 1 0
6 5 P1 2,3,4 1 0 0 0
7 6 P2 1,3,4 0 1 0 1
8 7 P1 2,3,4 1 0 1 1
9 8 P4 1,2,3 1 1 0 0
10 9 P1 2,3,4 0 0 1 0
11 10 P2 1,3,4 1 1 1 1
12 11 P1 2,3,4 0 0 0 1
13 12 P3 1,2,4 1 0 1 0
14 13 P1 2,3,4 0 1 0 0
15 14 P2 1,3,4 1 0 0 1
16 15 P1 2,3,4 0 1 1 1
Using the above algorithm, 4! i.e. 24 possible combinations of 4-bit binary inverse gray codes
can be generated.

236
3.2 Inverse Gray code for n- odd
Let us consider the construction of a 3-bit binary Inverse Gray code. All the integers, i.e.,
{0, 1, 2,3,…..(23
–1)} are arranged in the form of indicial sets as given below:
Q0 = 20
{1, 3, 5,7} = 1,3,5,7
Q1 = 21
{1, 3} = 2,6
Q2 = 22
{1} = 4
Since we are considering a 3-bit case, consider the permutation {2,3,1}. Hence, P1 = 2;
P2 = 3; P3 = 1. The first code word is (0 0 0) which is the zeroeth row of the code. To obtain 1st
row,
we have to change Pj
th
.
Here, 1 is in Q0. Therefore, all the other bits except P1 are
to be changed, hence the code is (1 0 1). Similarly, since ‘2’ is in Q1, all the other bits except P2 bit
are to be changed, hence the code is (1 1 0). This procedure is to be repeated for all the rows except
M/2 th row (in this case it is 4th
row).To obtain 4th
row all the 3-bits are to be changed. The resulting
code obtained by continuing this procedure is given in Table IV.
Table IV: 3-bit Inverse Gray Code for Permutation {2, 3, 1}
Element
No.
ith
row Pj Bits to be changed 3-bit Gray Code
3 2 1
1 0 - - 0 0 0
2 1 P1 1,3 1 0 1
3 2 P2 1,2 1 1 0
4 3 P1 1,3 0 1 1
5 4 All 1,2,3 1 0 0
6 5 P1 1,3 0 0 1
7 6 P2 1,2 0 1 0
8 7 P1 1,3 1 1 1
Similarly n! Inverse Gray Codes can be generated for any integer ‘n’ using n! permutations.
Table V gives Inverse Gray codes for n = 4 & 5 in decimal notation with different permutations.
Table V
n Permutation Inverse Gray code
4 3 1 2 4 0,11,5,14,3,8,6,13,10,1,15,4,9,2,12,7
5 3 4 1 2 5 0,27,12,23,9,18,5,30,3,24,15,20,10,1
7,6,29,2,25,14,21,11,16,7,28,1,26,13
,22,8,19,4,31

237
4. PARTIAL GENERATION OF 2n
-LENGTH WALSH CODES USING n-BIT GRAY AND
INVERSE GRAY CODES
Walsh code[1] is defined as a group of M code words which contain M binary elements
which with themselves and their logical inverses form a mutually orthogonal set. Two sequences are
said to be orthogonal when the Cross-correlation (inner product) between them is zero. Walsh Code
is known popularly as a group of spreading codes having good autocorrelation properties and poor
cross-correlation properties. Walsh codes are the backbone of CDMA systems and are used to
develop the individual channels in CDMA[3,5,6]. IS-95 uses 64-length Walsh Code set and these
allow the creation of 64 channels from the base station. Excluding the pilot and sync channels a base
station can talk to a maximum of 54 mobiles at the same time. CDMA 2000 uses 256-length Walsh
Code set. The method explained in this section allows us to construct only 2n words of 2n
-length
Walsh Code set using n-bit Gray and Inverse Gray Codes.
4.1. 8-length Walsh Code set Construction using 3-bit Gray and Inverse Gray Codes
With this construction procedure only 6 Walsh code words of length 8 can be obtained. A
combined 6-bit (3+3) code of 8(23
) code words is constructed by appending 3-bit Gray Code with
3-bit Inverse Gray Code generated using the algorithms of section II & III for a given permutation.
Each column of this combined code is a 8-length Walsh code word. Table VI gives the 8-length
Walsh Code set generated using the permutation {1, 3, 2}. 8-length Walsh Code word 2 (W2) is
bold-faced in column three of Table-VI.
4.2. 16-length Walsh Code set Construction using 4-bit Gray and Inverse Gray Codes
Similarly 8, Walsh code words of length-16 can be obtained from 8 columns of the combined
8-bit code constructed by pre-pending 4-bit Gray Code to the 4-bit Inverse Gray Code. Walsh code
word 1 (W1) of length 16 is high-lighted in column 3 of Table-VII. Here, it is observed that the
columns in the top-half of this combined code form a complete 8-length Walsh Code set (W0-W7).
The obtained Walsh code words with n=4 are tabulated in Table-VII. Column five of Table-VII gives
the complete 8-length Walsh Code set (W0-W7) generated using the procedure explained above. n!
Gray and Inverse Gray Codes result in n! different sequence orderings of constructed Walsh code
words.
This procedure can be extended to any value of ‘n’. Hence, it is concluded that by appending
n-bit Gray codes with n-bit Inverse Gray codes 2n Walsh code words of length 2n
can be obtained
from the 2n columns and also 2n Walsh code words of length 2n-1
can be obtained from the columns
in the top-half.
Table – VI
3-bit
Gray Code
3-bit Inverse
Gray Code
Combined
6-bit code
8-length Walsh
Code words
000
001
101
100
110
111
011
010
000
110
101
011
100
010
001
111
000000
001110
101101
100011
110100
111010
011001
010111
W2
W1
W4
W5
W7
W3

238
Table – VII
4-bit
Gray Code
4-bit Inverse
Gray Code
Combined
8-bit code
16-length
Walsh
Code words
8-length
Walsh Code
set
0000
0001
0011
0010
0110
0111
0101
0100
1100
1101
1111
1110
1010
1011
1001
1000
0000
1110
0011
1101
0110
1000
0101
1011
1100
0010
1111
0001
1010
0100
1001
0111
00000000
00011110
00110011
00101101
01100110
01111000
01010101
01001011
11001100
11010010
11111111
11100001
10101010
10110100
10011001
10000111
W1
W2
W4
W8
W14
W13
W11
W7
W0
W1
W2
W4
W7
W6
W5
W3
With n! Permutations n! Sequence orderings of 2n Walsh code words of length 2n
and 2n-1
are
obtained. The logical inverses of the generated Walsh code words in decimal notation for n = 3, 4, 5
with different permutations are given in Table-VIII.
Table-VIII
With permutation
{2,3,1}
8-length
Walsh Code
words
With permutation {3,1,2,4} With permutation {3,4,1,2,5}
16-length
Walsh
Code words
8-length
Walsh
Code words
32-length
Walsh Code
words
16-length
Walsh
Code words
153
240
195
204
170
150
65280
39321
61455
50115
43605
52428
42330
38550
255
153
240
195
170
204
165
150
4294901760
3284386755
2576980377
4278190335
4027576335
2863311530
2526451350
3435973836
2857719210
2774181210
65535
50115
39321
65280
61455
43690
38550
52428
43605
42330

239
5. CONCLUSION
2n Walsh Code words of length 2n
can be constructed using the proposed technique. This
technique allows us to construct 2n Walsh Code words of length 2n-1
. n! Sequence orderings can be
obtained with n! permutations, since they are constructed from n-bit Gray and Inverse Gray Codes.
With n=4, the columns in the top-half give a complete 8-length Walsh code set (n + n = 2n-1
). The
proposed algorithm is limited to 2n Walsh Code words. Possibility of complete Walsh Code set
generation using any value of ‘n’ is to be investigated. Future work includes the generation of
different Walsh Code sets of same length.
REFERENCES
[1] Walsh J.L., “A Closed set of Normal Orthogonal Functions”, American Journal of
Mathematics, 45, 1923.
[2] Gray F., “Pulse Code Communication”, US patent #2632058, 17th
March, 1953.
[3] K. J. Sankar, V.M. Pandharipande & P.S. Moharir, “Generalized Gray Codes”, Proc. Of IEEE
Intelligent Signal Processing and Communication Systems (ISPACS’04), 2004, 654–659.
[4] Duc-Minh Pham, A.B. Premkumar & A.S. Madhukumar, “Error Detection and Correction in
Communication Channels Using Inverse Gray RSNS codes”, IEEE Transactions on
communications, 59(4), 2011, 975-986.
[5] Usha K & Jaya sankar K, “ A Technique for the construction of Inverse Gray codes”,
International Journal of Emerging Trends in Engineering and Development, 2(2), 2012, 337-
341.
[6] Usha K & Jaya sankar K, “Generation of Walsh codes in two different orderings using 4-bit
Gray and Inverse Gray codes”, Indian Journal of Science and Technology, 5(3), 2012, 2341-
2345.
[7] Harmuth, H. F., "Applications of Walsh Functions in Communications", IEEE Spectrum 6,
1969, 82-91.
[8] Beauchamp, K. G., “Walsh Functions and Their Applications”. London: Academic Press,
1975.
[9] Prof. B.M. Mohan and Sanjeeb Kumar Kar, “Optimal Control of Multi-Delay Systems Via
Orthogonal Functions”, International Journal of Advanced Research in Engineering &
Technology (IJARET), Volume 1, Issue 1, 2010, pp. 1 - 24, ISSN Print: 0976-6480,
ISSN Online: 0976-6499.

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