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In this paper motivated from subspace coding we introduce subspace-metric and subset-metric codes. These are coordinate-position independent pseudometrics and suitable for the folded codes introduced by Guruswami and Rudra. The half-Singleton upper bounds for linear subspace-metric and subset-metric codes are proved. Subspace distances and subset distances of codes are natural lower bounds for insdel distances of codes, and then can be used to lower bound the insertion-deletion error-correcting capabilities of codes. The problem to construct efficient insertion-deletion error-correcting codes is notorious difficult and has attracted a long-time continuous efforts. The recent breakthrough is the algorithmic construction of near-Singleton optimal rate-distance tradeoff insertion-deletion code families by B. Haeupler and A. Shahrasbi in 2017 from their synchronization string technique. However most nice codes in these recent results are not explicit though many of them can be constructed by highly efficient algorithms. Our subspace-metric and subset-metric codes can be used to construct systemic explicit well-structured insertion-deletion codes. We present some near-optimal subspace-metric and subset-metric codes from known constant dimension subspace codes. By analysing the subset distances of folded codes from evaluation codes of linear mappings, we prove that they have high subset distances and then are explicit good insertion-deletion codes
This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n times n$ arrays affected by $t$-criss-cross deletions defined as any combination of $t_r$ ro
In this paper, we present an optimal metric function on average, which leads to a significantly low decoding computation while maintaining the superiority of the polarization-adjusted convolutional (PAC) codes error-correction performance. With our p
We derive simplified sphere-packing and Gilbert--Varshamov bounds for codes in the sum-rank metric, which can be computed more efficiently than previous ones. They give rise to asymptotic bounds that cover the asymptotic setting that has not yet been
This paper investigates the theory of sum-rank metric codes for which the individual matrix blocks may have different sizes. Various bounds on the cardinality of a code are derived, along with their asymptotic extensions. The duality theory of sum-ra
This paper extends the study of rank-metric codes in extension fields $mathbb{L}$ equipped with an arbitrary Galois group $G = mathrm{Gal}(mathbb{L}/mathbb{K})$. We propose a framework for studying these codes as subspaces of the group algebra $mathb