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

Abstract: We resolve an open problem posed by Chen--Cheng--Qi (IEEE Trans.\ Inf.\ Theory, 2025): can 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 $d_1\ge\tfrac{2}{3}d_{\mathrm{sr}}$ that underlies 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 of $C_1$.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.