Three-weight codes, triple sum sets, and strongly walk regular graphs

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

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“…The columns of its check matrix form a triple sum set in the sense of [16], since the sum of the weights (4 + 6 + 8) equals 3 × 8 × 3 4 .…”

Section: Since D Is An Integer This Yields D ≤ Nmentioning

confidence: 99%

Type IV codes over a non-unital ring

et al. 2021

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“…Despite of the number of the algebraic constructions for three-weight codes, see e.g. [13,14,15,33,36], there are few known geometric constructions [19].…”

Section: Some Known Linear Codes With Few Weightsmentioning

confidence: 99%

“…If ≤ r we say that the code has few weights. Much of the focus on linear codes to date has been on codes with few weights, especially on two and three-weight codes, for their applications in secret sharing [13], authentication codes [18] and their connections with association schemes [6,5] and with graphs [33].…”

Section: Introductionmentioning

confidence: 99%

Codes with few weights arising from linear sets

Napolitano¹,

Zullo²

2023

AMC

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“…Such sets, when the prescribed family of the subspaces is that of the hyperplanes and the range of the size of their intersections with hyperplanes is small, are related with other combinatorial objects such as for example graphs, some classes of difference sets and linear codes with few weights, see e.g. [8,9,17,18,35]. When considering this last relationship, to which we are interested in this paper, the existence problem takes on a special role.…”

Section: Introductionmentioning

confidence: 99%

Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

Napolitano¹,

Polverino²,

Santonastaso³

et al. 2023

AMC

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In this paper we consider two pointsets in <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{PG}(2,q^n) $\end{document}</tex-math></inline-formula> arising from a linear set <inline-formula><tex-math id="M3">\begin{document}$ L $\end{document}</tex-math></inline-formula> of rank <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> contained in a line of <inline-formula><tex-math id="M5">\begin{document}$ \mathrm{PG}(2,q^n) $\end{document}</tex-math></inline-formula>: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set <inline-formula><tex-math id="M6">\begin{document}$ L $\end{document}</tex-math></inline-formula>. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing <inline-formula><tex-math id="M7">\begin{document}$ L $\end{document}</tex-math></inline-formula> to be an <inline-formula><tex-math id="M8">\begin{document}$ {\mathbb F}_{q} $\end{document}</tex-math></inline-formula>-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the <inline-formula><tex-math id="M9">\begin{document}$ \Gamma\mathrm{L} $\end{document}</tex-math></inline-formula>-class of <inline-formula><tex-math id="M10">\begin{document}$ L $\end{document}</tex-math></inline-formula> and the number of inequivalent codes we can construct starting from it.

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Three-weight codes, triple sum sets, and strongly walk regular graphs

Cited by 22 publications

References 19 publications

Type IV codes over a non-unital ring

Type IV codes over a non-unital ring

Codes with few weights arising from linear sets

Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

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