arXiv subjects

Alessandro Neri

Publications and source records attributed to Alessandro Neri.

49 records · Page 3Linked to original sources

arXiv2020

Roos bound for skew cyclic codes in Hamming and rank metric

In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using the skew Roos bound, is also given. Moreover, some examples of MDS codes and MRD codes over finite fields are built, using the skew Roos bound.

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arXiv2020

Systematic encoders for generalized Gabidulin codes and the $q$-analogue of Cauchy matrices

We characterize the generator matrix in standard form of generalized Gabidulin codes. The parametrization we get for the non-systematic part of this matrix coincides with the $q$-analogue of generalized Cauchy matrices, leading to the definition of $q$-Cauchy matrices. These matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized Gabidulin codes whose generator matrix is a structured matrix. In particular, as an application, we construct Gabidulin codes whose generator matrix is the concatenation of an identity block and a Toeplitz/Hankel matrix. In addition, our results allow to give a new efficient criterion to verify whether a rank metric code of dimension $k$ and length $n$ is a generalized Gabidulin code. This criterion is only based on the computation of the rank of one matrix and on the verification of the linear independence of two sets of elements and it requires $\mathcal O(m\cdot F(k,n))$ field operations, where $F(k,n)$ is the cost of computing the reduced row echelon form of a $k \times n$ matrix. Moreover, we also provide a characterization of the generator matrix in standard form of general MRD codes.

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arXiv2019

A geometric characterization of minimal codes and their asymptotic performance

In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes.

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arXiv2019

Invariants and Inequivalence of Linear Rank-Metric Codes

We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. These invariants give rise to an easily computable criterion to check if two codes are inequivalent. With this criterion we then derive bounds on the number of equivalence classes of classical and twisted Gabidulin codes.

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arXiv2019

Tensor Representation of Rank-Metric Codes

We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor rank of a code to be the tensor rank of its generating tensors, and propose that this quantity is a significant coding theoretic parameter. By a result on the tensor rank of Kruskal from the 1970s, the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k+d-1$. We call codes that meet this bound minimal tensor rank (MTR) codes. It is known from results in algebraic complexity theory that an MTR code implies the existence of an MDS code. In this paper, we also address the converse problem, that of the existence of an MTR code, given an MDS code. We identify several parameters for which the converse holds and give explicit constructions of MTR codes using MDS codes. We furthermore define generalized tensor ranks, which give a refinement of the tensor rank as a code invariant. Moreover, we use these to distinguish inequivalent rank-metric codes.

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arXiv2019

New Lower Bounds for Permutation Codes using Linear Block Codes

In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an $[n,k,d]_q$ linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree $n$, having minimum distance at least $d$ and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code.

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arXiv2018

How many weights can a cyclic code have ?

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), {whose nonzero codewords have period equal to the length of the code}. Asymptotics are derived on the function $Γ(k,q),$ {that is defined as} the largest number of nonzero weights a cyclic code of dimension $k$ over $\F_q$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or $q$-ary, as well as in the $q$-ary Hamming code are determined, two difficult results of independent interest.

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arXiv2018

Maximum Weight Spectrum Codes

In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field $\F_q$ was introduced. In that work the authors studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero distinct Hamming weights, showing that its cardinality is upper bounded by $\frac{q^k-1}{q-1}$. They showed that this bound was sharp in the case $ q=2 $, and in the case $ k=2 $. They conjectured that the bound is sharp for every prime power $ q $ and every positive integer $ k $. In this work quickly establish the truth of this conjecture. We provide two proofs, each employing different construction techniques. The first relies on the geometric view of linear codes as systems of projective points. The second approach is purely algebraic. We establish some lower bounds on the length of codes that satisfy the conjecture, and the length of the new codes constructed here are discussed.

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arXiv2018

Random Construction of Partial MDS Codes

This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage system. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any maximum rank distance code. Then we define a standard form for the generator matrices of PMDS codes and use this form to give an algebraic description of PMDS generator matrices. This implies that over a sufficiently large finite field a randomly chosen generator matrix in PMDS standard form generates a PMDS code with high probability. This also provides sufficient conditions on the field size for the existence of PMDS codes.

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arXiv2017

Hartmann--Tzeng bound and Skew Cyclic Codes of Designed Hamming Distance

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann-Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports.

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arXiv2017

A Complete Classification of Partial-MDS (Maximally Recoverable) Codes with One Global Parity

Partial-MDS (PMDS) codes are a family of locally repairable codes, mainly used for distributed storage. They are defined to be able to correct any pattern of $s$ additional erasures, after a given number of erasures per locality group have occurred. This makes them also maximally recoverable (MR) codes, another class of locally repairable codes. It is known that MR codes in general, and PMDS codes in particular, exist for any set of parameters, if the field size is large enough. Moreover, some explicit constructions of PMDS codes are known, mostly (but not always) with a strong restriction on the number of erasures that can be corrected per locality group. In this paper we generalize the notion of PMDS codes to allow locality groups of different sizes. We give a general construction of such PMDS codes with $s=1$ global parity, i.e., one additional erasure can be corrected. Furthermore, we show that all PMDS codes for the given parameters are of this form, i.e., we give a classification of these codes. This implies a necessary and sufficient condition on the underlying field size for the existence of these codes (assuming that the MDS conjecture is true). For some parameter sets our generalized construction gives rise to PMDS codes with a smaller field size than any other known construction.

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arXiv2017

Fuzzy Authentication using Rank Distance

Fuzzy authentication allows authentication based on the fuzzy matching of two objects, for example based on the similarity of two strings in the Hamming metric, or on the similiarity of two sets in the set difference metric. Aim of this paper is to show other models and algorithms of secure fuzzy authentication, which can be performed using the rank metric. A few schemes are presented which can then be applied in different scenarios and applications.

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arXiv2016

On the Genericity of Maximum Rank Distance and Gabidulin Codes

We consider linear rank-metric codes in $\mathbb F_{q^m}^n$. We show that the properties of being MRD (maximum rank distance) and non-Gabidulin are generic over the algebraic closure of the underlying field, which implies that over a large extension field a randomly chosen generator matrix generates an MRD and a non-Gabidulin code with high probability. Moreover, we give upper bounds on the respective probabilities in dependence on the extension degree $m$.

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