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Bounds and Genericity of Sum-Rank-Metric Codes [article]

Cornelia Ott, Sven Puchinger, Martin Bossert
2021 arXiv   pre-print
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.  ...  We also provide two genericity results: we show that random linear codes achieve almost the sum-rank-metric Gilbert--Varshamov bound with high probability.  ...  bound for sum-rank metric codes.  ... 
arXiv:2102.02244v2 fatcat:576usv22wnfgdd7kt34lv2jgqm

Column Rank Distances of Rank Metric Convolutional Codes [chapter]

Diego Napp, Raquel Pinto, Joachim Rosenthal, Filipa Santana
2017 Lecture Notes in Computer Science  
We study and introduce the notion of column rank distance of rank metric convolutional codes for any given rate and finite field.  ...  Within this new framework we generalize previous results on column distances of Hamming and rank metric convolutional codes [3, 8] .  ...  The sum rank distance of a rank metric convolutional code C is defined as Next theorem establishes a bound on the sum rank distance of a rank metric convolutional code.  ... 
doi:10.1007/978-3-319-66278-7_21 fatcat:i2rnsc2wz5e6pdcsz44s4do4l4

New Explicit Good Linear Sum-Rank-Metric Codes [article]

Hao Chen
2022 arXiv   pre-print
Fundamental properties of sum-rank-metric codes have been studied and some explicit or probabilistic constructions of good sum-rank-metric codes have been proposed.  ...  Asymptotically our constructed sum-rank-metric codes are closing to the Gilbert-Varshamov-like bound for the sum-rank-metric codes for some parameters.  ...  Hence the sum-rank-metric is a generalization and combination of the Hamming metric and the rank-metric. The Singleton upper bound for the rank-sum-metric was proved in [2, 15] .  ... 
arXiv:2205.13087v3 fatcat:62zxoog3uzdjtmr3wnobuavhry

Fundamental Properties of Sum-Rank Metric Codes [article]

Eimear Byrne, Heide Gluesing-Luerssen, Alberto Ravagnani
2020 arXiv   pre-print
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  ...  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  ...  We have seen that the theory of sum-rank metric codes touches aspects of both rank-metric and Hamming-metric codes. Our approach has led us to various new bounds for sum-rank metric codes.  ... 
arXiv:2010.02779v1 fatcat:skvn7zcuivexfda3rlbkh7fgcm

On (Partial) Unit Memory Codes Based on Gabidulin Codes [article]

Antonia Wachter, Vladimir Sidorenko, Martin Bossert, Victor Zyablov
2011 arXiv   pre-print
Upper bounds for the free rank distance and the slope of (P)UM codes in the sum rank metric are derived and an explicit construction of (P)UM codes based on Gabidulin codes is given, achieving the upper  ...  For the sum rank metric, the free rank distance, the extended row rank distance and its slope are defined analogous to the extended row distance in Hamming metric.  ...  We defined general distance measures for convolutional codes based on a modified rank metric -the sum rank metric -and derived upper bounds for (P)UM codes based on the sum rank metric.  ... 
arXiv:1102.2700v2 fatcat:uzzc7shuznclrays7gfto7u5p4

Optimal anticodes, MSRD codes, and generalized weights in the sum-rank metric [article]

Eduardo Camps Moreno, Elisa Gorla, Cristina Landolina, Elisa Lorenzo García, Umberto Martínez-Peñas, Flavio Salizzoni
2021 arXiv   pre-print
Mathematically, the sum-rank metric is a natural generalization of both the Hamming metric and the rank metric.  ...  ., codes attaining the sum-rank metric Anticode Bound. We use these optimal anticodes to define generalized sum-rank weights and we study their main properties.  ...  Using these results, we provide in Section 3 our Anticode Bound for sum-rank metric codes and we provide an explicit description and classification of optimal anticodes for the sum-rank metric.  ... 
arXiv:2107.13611v2 fatcat:7pvwwqzptzf7laenf7gwlgare4

Codes in the Sum-Rank Metric: Fundamentals and Applications

Umberto Martínez-Peñas, Mohannad Shehadeh, Frank R. Kschischang
2022 Foundations and Trends in Communications and Information Theory  
Other families of codes in the sum-rank metric, including convolutional codes and subfield subcodes are described, and recent results in the general theory of codes in the sum-rank metric are surveyed.  ...  This monograph provides a tutorial introduction to the theory and applications of sum-rank metric codes over finite fields.  ...  Apart from its role in applications, the sum-rank metric (1.3) can be seen as a generalization of the Hamming and rank metrics.  ... 
doi:10.1561/0100000120 fatcat:egfyeckaofdhld7wpns7qreyzy

Theory of supports for linear codes endowed with the sum-rank metric [article]

Umberto Martínez-Peñas
2019 arXiv   pre-print
to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found.  ...  The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields.  ...  bounds on generalized Hamming and rank weights [35] ; 2) We connect the MSRD property with sum-rank supports, and prove that duals, shortenings and restrictions of MSRD codes give again MSRD codes; 3  ... 
arXiv:1808.02468v2 fatcat:lvyrfu2llffyleabgumzjcyxoa

Some Matroids Related to Sum-Rank Metric Codes [article]

Avijit Panja, Rakhi Pratihar, Tovohery Hajatiana Randrianarisoa
2022 arXiv   pre-print
We introduce the notion of sum-matroids and show its association with sum-rank metric codes.  ...  As a consequence, some results for sum-rank metric codes by Martínez-Peñas are generalized for sum-matroids. The sum-matroids generalize the notions of matroids and q-matroids.  ...  Sudhir Ghorpade for reading our work and giving many useful comments that helped us to make approapriate revisions in this version.  ... 
arXiv:1912.09984v4 fatcat:ftshc6gfqnh2potftalaoz52oq

MRD rank metric convolutional codes

Diego Napp, Raquel Pinto, Joachim Rosenthal, Paolo Vettori
2017 2017 IEEE International Symposium on Information Theory (ISIT)  
We introduce a rank distance, which is suitable for convolutional codes, and derive a new Singleton-like upper bound.  ...  In this paper, we propose rank metric convolutional codes for this purpose. The framework we present is slightly more general than the one which can be found in the literature.  ...  Specifically, after recalling some basic facts about convolutional and rank metric codes, we introduce a first general definition of rank metric convolutional codes, we propose a suitable concept of distance  ... 
doi:10.1109/isit.2017.8007033 dblp:conf/isit/NappPRV17 fatcat:oe5t4mqevfdkta4yecppoza7pa

List-decodable Codes and Covering Codes [article]

Hao Chen
2022 arXiv   pre-print
Some new better results about non-list-decodability of rank-metric codes, subspace codes, sum-rank-metric codes and permutation codes with various metrics are obtained.  ...  We apply our general covering code upper bounds for list-decodable rank-metric codes, list-decodable subspace codes, list-decodable insertion codes list-decodable deletion codes,list-decodable sum-rank-metric  ...  When n = 1, the Hamming metric is recovered. When t = 1, the rank-metric is recovered. Hence the sum-rank-metric is a generalization and combination of the Hamming metric and the rank-metric.  ... 
arXiv:2109.02818v13 fatcat:7zmmcst44nbfjixv7u2dhyppii

Hamming and simplex codes for the sum-rank metric [article]

Umberto Martínez-Peñas
2021 arXiv   pre-print
General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters.  ...  Bounds on the minimum sum-rank distance of sum-rank simplex codes are given based on known bounds on the size of partial spreads.  ...  For this reason, and since Reed-Solomon codes and Hamming codes are not equal (for the Hamming metric), we do not expect linearized Reed-Solomon codes to be sum-rank Hamming codes in general.  ... 
arXiv:1908.03239v2 fatcat:5vopp3cknrgnln74mrb7ayooaa

A Singleton Bound for Generalized Ferrers Diagram Rank Metric Codes [article]

Srikanth B. Pai, B. Sundar Rajan
2015 arXiv   pre-print
In this paper, we will employ the technique used in the proof of classical Singleton bound to derive upper bounds for rank metric codes and Ferrers diagram rank metric codes.  ...  A new Singleton bound for generalized Ferrers diagram rank metric code is obtained using our technique.  ...  Singleton Bound for Generalized Ferrers Diagram Rank Metric Codes We will prove a Singleton bound for generalized Ferrers diagram rank metric code and obtain the Singleton bound for Ferrers diagram rank  ... 
arXiv:1506.05558v1 fatcat:z6rtkf7xmrdk7eujrwovwzi2km

Bounds on List Decoding of Linearized Reed-Solomon Codes [article]

Sven Puchinger, Johan Rosenkilde
2021 arXiv   pre-print
In the two extreme cases of the sum-rank metric, they coincide with Reed-Solomon codes (Hamming metric) and Gabidulin codes (rank metric).  ...  Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality.  ...  The class of linearized Reed-Solomon (LRS) codes were introduced in [10] , and for a given sum-rank metric, there is a large family of LRS codes meeting the Singleton bound for that metric.  ... 
arXiv:2102.03079v1 fatcat:3jyd2rfiivbnbhyu34fq6dc3mq

Bounds on List Decoding of Rank-Metric Codes

Antonia Wachter-Zeh
2013 IEEE Transactions on Information Theory  
Second, an exponential upper bound is derived, which holds for any rank-metric code of length n and minimum rank distance d.  ...  All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.  ...  2 Problem Statement 3 A Bound on the List Size of Gabidulin Codes Lower Bound Asymptotic Behavior of Bounds 4 Bounds for General Rank-Metric Codes Interpretation as Constant-Rank Code Upper  ... 
doi:10.1109/tit.2013.2274653 fatcat:x2suzt4f3rbwvje4oldut6t2nm
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