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Register a new userExplicit Good Subspace-metric and Subset-metric Codes as Insertion-deletion Codes
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Added by
Hao Chen
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Hao Chen
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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
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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 $t_r$ row and $t_c$ column deletions such that $t_r + t_c = t$ for a given $t$. We show an equivalence between correcting $t$-criss-cross deletions and $t$-criss-cross insertions and show that a code correcting $t$-criss-cross insertions/deletions has redundancy at least $tn + t log n - log(t!)$. Then, we present an existential construction of $t$-criss-cross insertion/deletion correcting code with redundancy bounded from above by $tn + mathcal{O}(t^2 log^2 n)$. The main ingredients of the presented code construction are systematic binary $t$-deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient.
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 proposed metric function, the PAC codes decoding computation is comparable to the conventional convolutional codes (CC) sequential decoding. Moreover, simulation results show an improvement in the low-rate PAC codes error-correction performance when using our proposed metric function. We prove that choosing the polarized cutoff rate as the metric functions bias value reduces the probability of the sequential decoder advancing in the wrong path exponentially with respect to the wrong path depth. We also prove that the upper bound of the PAC codes computation has a Pareto distribution; our simulation results also verify this. Furthermore, we present a scaling-bias procedure and a method of choosing threshold spacing for the search-limited sequential decoding that substantially improves the decoders average computation. Our results show that for some codes with a length of 128, the search-limited PAC codes can achieve an error-correction performance close to the error-correction performance of the polar codes under successive cancellation list decoding with a list size of 64 and CRC length of 11 with a considerably lower computation.
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 considered in the literature: families of sum-rank-metric codes whose block size grows in the code length. We also provide two genericity results: we show that random linear codes achieve almost the sum-rank-metric Gilbert--Varshamov bound with high probability. Furthermore, we derive bounds on the probability that a random linear code attains the sum-rank-metric Singleton bound, showing that for large enough extension fields, almost all linear codes achieve it.
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-rank metric codes is also explored, showing that MSRD codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all matrix blocks have the same number of columns. In the latter case, duality considerations lead to an upper bound on the number of blocks for MSRD codes. The paper also contains various constructions of sum-rank metric codes for variable block sizes, illustrating the possible behaviours of these objects with respect to bounds, existence, and duality properties.
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 $mathbb{L}[G]$, and we relate this point of view with usual notions of rank-metric codes in $mathbb{L}^N$ or in $mathbb{K}^{Ntimes N}$, where $N = [mathbb{L} : mathbb{K}]$. We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes. We then focus on the case where $G$ is abelian, which leads us to see codewords as elements of a multivariate skew polynomial ring. We prove that we can bound the dimension of the vector space of zeroes of these polynomials, depending of their degree. This result can be seen as an analogue of Alon-Furedi theorem -- and by means, of Schwartz-Zippel lemma -- in the rank metric. Finally, we construct the counterparts of Reed-Muller codes in the rank metric, and we give their parameters. We also show the connection between these codes and classical Reed-Muller codes in the case where $mathbb{L}$ is a Kummer extension.
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