An Algebraic Approach for Decoding Spread Codes

TL;DR

This paper introduces an algebraic decoding algorithm for spread codes, a class of optimal constant-dimension codes used in network coding, which improves efficiency for small code dimensions by leveraging algebraic structures.

Contribution

The paper presents a novel algebraic decoding algorithm for spread codes that is more efficient for small code dimensions than previous methods.

Findings

Decoding complexity is O((n-k)k^3) over an extension field.

Algorithm exploits algebraic structure and polynomial factorization.

Improves decoding efficiency for small code dimensions.

Abstract

In this paper we study spread codes: a family of constant-dimension codes for random linear network coding. In other words, the codewords are full-rank matrices of size (k x n) with entries in a finite field F_q. Spread codes are a family of optimal codes with maximal minimum distance. We give a minimum-distance decoding algorithm which requires O((n-k)k^3) operations over an extension field F_{q^k}. Our algorithm is more efficient than the previous ones in the literature, when the dimension k of the codewords is small with respect to n. The decoding algorithm takes advantage of the algebraic structure of the code, and it uses original results on minors of a matrix and on the factorization of polynomials over finite fields.