Abstract
Cyclically permutable codes (CPCs) are important to communication networks, e.g., multiple access collision channels without feedback and frequency-hopping spread spectrum communication channels. A CPC is a block code of length n such that each codeword has full cyclic order n and all codewords are cyclically distinct. This study investigate the characteristics of finite fields to develop an efficient algorithm to find a CPC from a p-ary cyclic code, where p is a prime number. In this paper, the Galois field Fourier transform technique is used to generate a CPC of either primitive or non-primitive length.
Highlights
For the past few years, cyclically permutable codes and their applications in communication networks, e.g. multiple access collision channel without feedback (1) and frequencyhopping spread spectrum communications channels (2), (3), and digital watermarking (4), (5) have become increasingly important
Gilbert (7) defined a cyclically permutable code (CPC) as a block code of length n, such that each codeword has cyclic order $n$ and any cyclic shift of all codewords are distinct, i.e., no codeword in Cyclically permutable codes (CPCs) can be obtained by any cyclic shift of another codeword
As opposed to (11), which used the time-domain to find a CPC, this study proposes the use of the frequency domain as an efficient method to find many CPCs from cyclic codes
Summary
For the past few years, cyclically permutable codes and their applications in communication networks, e.g. multiple access collision channel without feedback (1) and frequencyhopping spread spectrum communications channels (2), (3), and digital watermarking (4), (5) have become increasingly important. Gilbert (7) defined a cyclically permutable code (CPC) as a block code of length n, such that each codeword has cyclic order $n$ and any cyclic shift of all codewords are distinct, i.e., no codeword in CPC can be obtained by any cyclic shift of another codeword. In (11), the authors proposed the use of algebraic property of a binary cyclic code, such as the generator polynomial in timedomain, for an efficient and systematic construction of a CPC from this binary cyclic code. We use the Galois field Fourier transform method to form a CPC from a p-ary cyclic code of length n where n is a divisor of pm-1.
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