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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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {I}_{\mathcal {A}}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\mathcal {R}_{\mathcal {A}}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}_{\mathcal {A}}$ </tex-math></inline-formula>. The relationship between these triplets was studied for codewords that are cyclic shifts of each other. We introduce a set <inline-formula> <tex-math notation="LaTeX">$\mathcal {S}(\mathcal {C})$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {S}(\mathcal {C})$ </tex-math></inline-formula> 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.

Novel Indexing of Cyclic Codes With Run-Length Applications