Two-Step Decoding of Binary $2\times2$ Sum-Rank-Metric Codes

TL;DR

This paper introduces a two-step decoding method for binary sum-rank-metric codes that reduces decoding to constituent Hamming codes without restrictive conditions, achieving optimal complexity.

Contribution

It demonstrates that decoding sum-rank-metric codes can be simplified to decoding constituent Hamming codes without the previously required constraints.

Findings

Decoding can be reduced to constituent Hamming decoders without the $d_1 \\ge \\frac{2}{3}d_{sr}$ condition.

The proposed decoder achieves unique decoding up to half the minimum sum-rank distance.

Decoding complexity is $O(\\ell^2)$ for BCH or Goppa codes over \\F_4$.

Abstract

We address an open problem posed by Chen-Cheng-Qi (IEEE Trans.\ Inf.\ Theory, 2025): can the decoding of binary sum-rank-metric codes $\SR(C_1,C_2)$ with $2\times2$ matrix blocks be reduced entirely to decoding the constituent Hamming-metric codes $C_1$ and $C_2$ without the additional requirement $d_1\ge\tfrac{2}{3}d_{\mathrm{sr}}$ used in their fast decoder? We answer this in the affirmative by exhibiting a simple two-step procedure: first uniquely decode $C_2$, then apply a single error-erasure decoding for $C_1$. This shows that the restrictive hypothesis $d_1\ge\tfrac{2}{3}d_{\mathrm{sr}}$ is theoretically unnecessary. The resulting decoder achieves unique decoding up to $\lfloor (d_{\mathrm{sr}}-1)/2\rfloor$ with overall cost $T_2+T_1$, where $T_2$ and $T_1$ are the complexities of the Hamming decoders for $C_2$ and $C_1$, respectively. We further show that this reduction is asymptotically optimal in a black-box model, as any sum-rank decoder must inherently decode the constituent Hamming codes. For BCH or Goppa instantiations over $\F_4$, the decoder runs in $O(\ell^2)$ time.