Novel Indexing of Cyclic Codes With Run-Length Applications

Abstract

Any cyclic code over a finite field can be considered as a sum of some non-degenerate irreducible cyclic codes. We use the traditional trace function representation of irreducible cyclic codes to suggest a similar representation to any cyclic code. This representation suggests an indexing for codewords of the cyclic code. Specifically, each codeword is identified by a triplet IA, RA and QA. The relationship between these triplets was studied for codewords that are cyclic shifts of each other. We introduce a set S(C) whose elements correspond to subsets that each contains all the cyclic shifts of a codeword. As an application to the proposed indexing, we take advantage of the trace function linearity to explore codewords with a specific run-length. Because the run-length is invariant to cyclic shifts, it suffices to explore elements of S(C) that match codewords with a specific run-length. Instead of searching at the code for these codewords, the problem is turned into solving a system of linear equations.