Algebraic List-decoding of Subspace Codes

TL;DR

This paper develops an algebraic list-decoding method for Koetter-Kschischang subspace codes, significantly improving error correction capabilities in network coding by generalizing existing constructions and achieving higher decoding radii.

Contribution

It introduces a novel algebraic list-decoding algorithm for subspace codes, extending Koetter-Kschischang codes and surpassing previous error correction bounds.

Findings

Achieves successful list-decoding up to a higher error radius than previous bounds.

Generalizes Koetter-Kschischang construction for improved decoding.

Provides theoretical guarantees for decoding performance at various rates.

Abstract

Subspace codes were introduced in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). Subspace codes are indeed collections of subspaces of a certain vector space over a finite field. The Koetter-Kschischang construction of subspace codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang subspace codes. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer $L$, our list-$L$ decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error is at most $ L - \frac{L(L+1)}{2}R $ where $R$ is the normalized packet rate. Just as in the case of Sudan's list-decoding algorithm, this exceeds the previously best known error-correction radius $1-R$, demonstrated by Koetter and Kschischang, for low rates $R$.