Decoding of Interleaved Alternant Codes

TL;DR

This paper extends decoding bounds for interleaved alternant codes, a class of subfield subcodes of Reed-Solomon codes, providing new theoretical limits on their error correction capabilities.

Contribution

It establishes new lower and upper bounds on the fraction of decodable error matrices for interleaved alternant codes using Schmidt et al.'s decoding algorithm.

Findings

New bounds improve understanding of decoding success rates.

Bounds are the first to be known for interleaved alternant codes.

Results enable better assessment of decoding reliability.

Abstract

Interleaved Reed-Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed-Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either fail to return a codeword or miscorrect to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed-Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.'s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known.