Further results on the classification of MDS codes

Kokkala, Janne I.; Östergård, Patric R. J.

doi:10.3934/amc.2016020

Cited by 9 publications

(13 citation statements)

References 20 publications

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“…See [3] for a recent survey. It is also known to hold for all MDS codes over alphabets of size at most 8, see [18]. Here we present some evidence that the MDS conjecture is true for additive MDS codes over finite fields by proving the conjecture for additive MDS codes over F 9 and F 16 , where in the last case we assume linearity over F 4 .…”

Section: Introductionmentioning

confidence: 61%

“…Thus, from Table 1, we conclude that all additive MDS codes over F 4 are equivalent to a linear code over F 4 . Indeed, it was already observed in [18], that all MDS codes over an alphabet of size 4 are equivalent to linear codes. This also follows from the Blokhuis-Brouwer theorem […”

Section: Additive Mds Codes Over Small Finite Fieldsmentioning

confidence: 99%

“…In [18], it was proven that all (n, 8 k , d) 8 MDS codes with d ≥ 5 are equivalent to linear codes and all (n, 8 k , 4) 8 MDS codes with k ≥ 4 are equivalent to linear codes. There are precisely 39 equivalence classes of (6, 8 3 , 4) 8 codes.…”

Section: Additive Mds Codes Over Small Finite Fieldsmentioning

confidence: 99%

“…The non-existence of two mutually orthogonal Latin squares of order 6 implies the non-existence of nontrivial MDS codes with d ≥ 3 over alphabets of size 6. In the articles [17] and [18], all MDS codes over alphabets of size 5, 7 and 8 are classified. It turns out that all MDS codes over alphabets of size 5 and 7 with d ≥ 3, except the (4, 7 2 , 3) 7 codes, are equivalent to linear MDS codes.…”

Section: Introductionmentioning

confidence: 99%

“…It turns out that all MDS codes over alphabets of size 5 and 7 with d ≥ 3, except the (4, 7 2 , 3) 7 codes, are equivalent to linear MDS codes. Here, for the sake of completeness, we also classify additive MDS codes over F 8 and compare this classification to the results obtained in [18].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On additive MDS codes over small fields

Ball¹,

Gamboa²,

Lavrauw³

2023

AMC

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ C $\end{document}</tex-math></inline-formula> be a <inline-formula><tex-math id="M2">\begin{document}$ (n,q^{2k},n-k+1)_{q^2} $\end{document}</tex-math></inline-formula> additive MDS code which is linear over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_q $\end{document}</tex-math></inline-formula>. We prove that if <inline-formula><tex-math id="M4">\begin{document}$ n \geq q+k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ k+1 $\end{document}</tex-math></inline-formula> of the projections of <inline-formula><tex-math id="M6">\begin{document}$ C $\end{document}</tex-math></inline-formula> are linear over <inline-formula><tex-math id="M7">\begin{document}$ {\mathbb F}_{q^2} $\end{document}</tex-math></inline-formula> then <inline-formula><tex-math id="M8">\begin{document}$ C $\end{document}</tex-math></inline-formula> is linear over <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb F}_{q^2} $\end{document}</tex-math></inline-formula>. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over <inline-formula><tex-math id="M10">\begin{document}$ {\mathbb F}_q $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M11">\begin{document}$ q \in \{4,8,9\} $\end{document}</tex-math></inline-formula>. We also classify the longest additive MDS codes over <inline-formula><tex-math id="M12">\begin{document}$ {\mathbb F}_{16} $\end{document}</tex-math></inline-formula> which are linear over <inline-formula><tex-math id="M13">\begin{document}$ {\mathbb F}_4 $\end{document}</tex-math></inline-formula>. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for <inline-formula><tex-math id="M14">\begin{document}$ q \in \{ 2,3\} $\end{document}</tex-math></inline-formula>.</p>

show abstract

Section: Introductionmentioning

confidence: 61%

Section: Additive Mds Codes Over Small Finite Fieldsmentioning

confidence: 99%

Section: Additive Mds Codes Over Small Finite Fieldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On additive MDS codes over small fields

Ball¹,

Gamboa²,

Lavrauw³

2023

AMC

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The equivalence of linear codes implies semi-linear equivalence

Ball

Dixon

2022

Des. Codes Cryptogr.

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Constructions of transitive latin hypercubes

Krotov

Potapov

2016

European Journal of Combinatorics

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A function f : {0, ..., q − 1} n → {0, ..., q − 1} invertible in each argument is called a latin hypercube. A collection (π 0 , π 1 , ..., π n ) of permutations of {0, ..., q − 1} is called an autotopism of a latin hypercube f if π 0 f (x 1 , ..., x n ) = f (π 1 x 1 , ..., π n x n ) for all x 1 , ..., x n . We call a latin hypercube isotopically transitive (topolinear) if its group of autotopisms acts transitively (regularly) on all q n collections of argument values. We prove that the number of nonequivalent topolinear latin hypercubes growths exponentially with respect to √ n if q is even and exponentially with respect to n 2 if q is divisible by a square. We show a connection of the class of isotopically transitive latin squares with the class of G-loops, known in noncommutative algebra, and establish the existence of a topolinear latin square that is not a group isotope. We characterize the class of isotopically transitive latin hypercubes of orders q = 4 and q = 5.

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Further results on the classification of MDS codes

Cited by 9 publications

References 20 publications

On additive MDS codes over small fields

On additive MDS codes over small fields

The equivalence of linear codes implies semi-linear equivalence

Constructions of transitive latin hypercubes

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