On the shape of the general error locator polynomial for cyclic codes

TL;DR

This paper provides explicit forms and sparsity analysis of the general error locator polynomial for cyclic codes, offering insights into decoding complexity for codes with up to three errors and lengths under 63.

Contribution

It presents a general explicit form for the error locator polynomial for all cyclic codes and proves its sparsity for codes with up to three errors and length less than 63.

Findings

Explicit form of the error locator polynomial for all cyclic codes.

Sparsity of the polynomial for codes with t ≤ 3 and n < 63.

Implications for decoding complexity of cyclic codes.

Abstract

A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for infinite classes of cyclic codes with $t=2$ and $t=3$. From these, a theoretically justification of the sparsity of the general error locator polynomial is obtained for all cyclic codes with $t\leq 3$ and $n<63$, except for three cases where the sparsity is proved by a computer check. Moreover, we discuss some consequences of our results to the understanding of the complexity of bounded-distance decoding of cyclic codes.