On the Decoding Complexity of Cyclic Codes Up to the BCH Bound

TL;DR

This paper improves the computational efficiency of algebraic decoding algorithms for cyclic codes up to the BCH bound, reducing complexity from linear to sublinear in certain cases, especially for binary codes.

Contribution

It introduces methods to significantly lower the decoding complexity for cyclic codes up to the BCH bound, particularly for binary codes, making decoding more practical for larger code lengths.

Findings

Syndrome computation complexity reduced from O(nt) to O(t√n) for binary codes.

Error location complexity reduced to at most max{O(t√n), O(t^2 log(t) log(n))}.

Decoding becomes more feasible for larger codes due to reduced computational requirements.

Abstract

The standard algebraic decoding algorithm of cyclic codes $[n,k,d]$ up to the BCH bound $t$ is very efficient and practical for relatively small $n$ while it becomes unpractical for large $n$ as its computational complexity is $O(nt)$. Aim of this paper is to show how to make this algebraic decoding computationally more efficient: in the case of binary codes, for example, the complexity of the syndrome computation drops from $O(nt)$ to $O(t\sqrt n)$, and that of the error location from $O(nt)$ to at most $\max \{O(t\sqrt n), O(t^2\log(t)\log(n))\}$.