Generation of Walsh codes in two different orderings using 4-bit Gray and Inverse Gray codes

Abstract

Walsh code sequences are fixed power codes and are widely used in multi-user CDMA communications. Walsh code is a group of spreading codes having good autocorrelation properties and poor cross-correlation properties. This paper presents a simple technique to construct Walsh code sets of any length recursively using 4-bit Gray and Inverse Gray codes. An n-bit Gray code is a list of all 2 n bit strings such that adjacent code words in the sequence differ in only one bit position. An 'n' bit Inverse Gray code , is defined exactly opposite to Gray code, it is a list of all 2 n bit strings of length 'n' each, such that successive code words differ in (n-1) bit positions. If the first and last code words also differ in one bit position then the resultant code is called cyclic. The technique presented in this paper allows us to construct 4! Walsh code set (of any length) orderings since they are constructed from 4-bit Gray and Inverse Gray codes. All these Walsh code sets are not symmetrical along rows and columns. A Gray-Binary mapping technique is adopted to transform these Walsh code sets into symmetrical matrices. n-bit Gray codes are used for mapping 2 n -length Walsh code sets. Out of n! permutations few result in equal row column transition counts. And two permutations transform these Walsh code sets into Walsh-Hadamard and Walsh-Paley sequence orderings.