Non-Linear Maximum Rank Distance Codes in the Cyclic Model for the Field Reduction of Finite Geometries

Durante, Nicola; Siciliano, Alessandro

doi:10.37236/6106

Cited by 21 publications

(25 citation statements)

References 28 publications

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“…Next we consider several possible cases of c i and d i satisfying (8) and (9). As we have proved that there are three elements in A, we only have to look at the following three cases.…”

Section: Similarly We Can Verify That a Is Neithermentioning

confidence: 99%

Generalized twisted Gabidulin codes

Lunardon

Trombetti

Zhou

2018

Journal of Combinatorial Theory, Series A

120

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Let C be a set of m by n matrices over Fq such that the rank of A − B is at least d for all distinct A, B ∈ C. Suppose that m n. If #C = q n(m−d+1) , then C is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary 1 < d < m − 1. One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016), and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family. q . When d(C) = d, it is well-known that #C ≤ q max{m,n}(min{m,n}−d+1) ,

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“…Next we consider several possible cases of c i and d i satisfying (8) and (9). As we have proved that there are three elements in A, we only have to look at the following three cases.…”

Section: Similarly We Can Verify That a Is Neithermentioning

confidence: 99%

Generalized twisted Gabidulin codes

Lunardon

Trombetti

Zhou

2018

Journal of Combinatorial Theory, Series A

120

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“…• Non-linear MRD codes by Cossidente, the second author and Pavese [5] which were later generalized by Durante and Siciliano [17]. • Linear MRD codes associated with maximum scattered linear sets of PG(1, q 6 ) and PG(1, q 8 ) presented recently in [1,7,9,38,54].…”

Section: Introductionmentioning

confidence: 99%

MRD codes with maximum idealizers

Csajbók

Marino

Polverino

et al. 2020

Discrete Mathematics

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Left and right idealizers are important invariants of linear rankdistance codes. In the case of maximum rank-distance (MRD for short) codes in F n×n q the idealizers have been proved to be isomorphic to finite fields of size at most q n. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in F n×n q for n ≤ 9 with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rankdistance codes providing MRD ones with maximum idealizers for n = 7, q odd and for n = 8, q ≡ 1 (mod 3). These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for n ≥ 9.

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“…For different relations between linear MRD codes and linear sets see [9,22], [30, Section 5], [7, Section 5]. To the extent of our knowledge, these are the only infinite families of linear MRD codes with m < n appearing in the literature.In [12] infinite families of non-linear (n, n, q; n − 1)-MRD codes, for q ≥ 3 and n ≥ 3 have been constructed. These families contain the non-linear MRD codes Aut(Ω m,n ) = {τ ∈ ΓL(Ω m,n ) : rank(f τ ) = rank(f ), for all f ∈ Ω m,n }.…”

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confidence: 99%

“…In [12] infinite families of non-linear (n, n, q; n − 1)-MRD codes, for q ≥ 3 and n ≥ 3 have been constructed. These families contain the non-linear MRD codes Aut(Ω m,n ) = {τ ∈ ΓL(Ω m,n ) : rank(f τ ) = rank(f ), for all f ∈ Ω m,n }.…”

mentioning

confidence: 99%

Puncturing maximum rank distance codes

Csajbók

Siciliano

2018

J Algebr Comb

Self Cite

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We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models we consider an infinite family of linear maximum rank distance codes obtained by puncturing generalized twisted Gabidulin codes. We calculate the automorphism group of such codes and we prove that this family contains many codes which are not equivalent to any generalized Gabidulin code. This solves a problem posed recently by Sheekey in [30].Recently, Sheekey [30] presented a new family of linear MRD codes by using linearized polynomials over F q n . These codes are now known as generalized twisted Gabidulin codes. The equivalence classes of these codes were determined by Lunardon, Trombetti and Zhou in [23]. In [28] a further generalization was considered giving new MRD codes when m < n; the authors call these codes generalized twisted Gabidulin codes as well. In this paper the term "generalized twisted Gabidulin code" will be used for codes defined in [30, Remark 8]. For different relations between linear MRD codes and linear sets see [9,22], [30, Section 5], [7, Section 5]. To the extent of our knowledge, these are the only infinite families of linear MRD codes with m < n appearing in the literature.In [12] infinite families of non-linear (n, n, q; n − 1)-MRD codes, for q ≥ 3 and n ≥ 3 have been constructed. These families contain the non-linear MRD codes Aut(Ω m,n ) = {τ ∈ ΓL(Ω m,n ) : rank(f τ ) = rank(f ), for all f ∈ Ω m,n }. By [36, Theorem 3.4],Aut(Ω m,n ) = (GL(V ) × GL(V ′ )) ⋊ Aut(F q ) for m < n,where ⊤ is an involutorial operator. In details, any given (g, g ′ ) ∈ GL(V ) × GL(V ′ ) defines the linear automorphism of Ω m,n given byfor any f ∈ Ω m,n . If A and B are the matrices of g ∈ GL(V ) and g ′ ∈ GL(V ′ ) in the given bases for V and V ′ , then the matrix of f (g,g ′ ) is A t M f B, where t denotes transposition. Additionally, the semilinear transformation φ of Ω m,n is the automorphism given by

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Non-Linear Maximum Rank Distance Codes in the Cyclic Model for the Field Reduction of Finite Geometries

Cited by 21 publications

References 28 publications

Generalized twisted Gabidulin codes

Generalized twisted Gabidulin codes

MRD codes with maximum idealizers

Puncturing maximum rank distance codes

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