Abstract
In the theory of cyclic codes, it is common practice to require that (n,q)=1, where n is the word length and F/sub q/ is the alphabet. It is shown that the even weight subcodes of the shortened binary Hamming codes form a sequence of repeated-root cyclic codes that are optimal. In nearly all other cases, one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime. This statement is based on an analysis for lengths up to 100. A theorem shows why this was to be expected, but it also leads to low-complexity decoding methods. This is an advantage, especially for the codes that are not much worse than corresponding codes of odd length. It is demonstrated that a binary cyclic code of length 2n (n odd) can be obtained from two cyclic codes of length n by the well-known mod u mod u+v mod construction. This leads to an infinite sequence of optimal cyclic codes with distance 4. Furthermore, it is shown that low-complexity decoding methods can be used for these codes. The structure theorem generalizes to other characteristics and to other lengths. Some comparisons of the methods using earlier examples are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.