Codes with few weights arising from linear sets

Vito Napolitano; Ferdinando Zullo

doi:10.3934/amc.2020129

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2023, Volume 17, Issue 2: 320-332. Doi: 10.3934/amc.2020129

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Codes with few weights arising from linear sets

Vito Napolitano and
Ferdinando Zullo^,

Dipartimento di Matematica e Fisica, , Università degli Studi della Campania "Luigi Vanvitelli", I– 81100 Caserta, Italy

^* Corresponding author: Ferdinando Zullo
^* Corresponding author: Ferdinando Zullo

Received: May 2020

Revised: October 2020

Early access: December 2020

Published: April 2023

This research was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). The authors were also supported by the project "VALERE: VAnviteLli pEr la RicErca" of the University of Campania "Luigi Vanvitelli"

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Abstract

In this article we present a class of codes with few weights arising from a special type of linear sets. We explicitly show the weights of such codes, their weight enumerators and possible choices for their generator matrices. In particular, our construction yields linear codes with three weights and, in some cases, almost MDS codes. The interest for these codes relies on their applications to authentication codes and secret schemes, and their connections with further objects such as association schemes and graphs.

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Mathematics Subject Classification: 51E20, 05B25, 51E22.

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Table 1. Possible choices for $ f_1, \ldots, f_r $

$ n $	$ r $	$ (f_1(x), \ldots, f_r(x)) $	conditions	references
		$ (x, x^{q^s}, \ldots, x^{q^{s(r-1)}}) $	$ \gcd(s, n)=1 $	[12,20,21]
		$ (x^{q^s}, \ldots, x^{q^{s(r-2)}}, x+\delta x^{q^{s(r-1)}}) $	$ \begin{array}{cc} \gcd(s, n)=1, \\ \mathrm{N}_{q^n/q}(\delta)\neq (-1)^{nr}\end{array} $	[31,25]
$ 6 $	$ 4 $	$ (x^q, x^{q^2}, x^{q^4}, x-\delta^{q^5} x^{q^{3}}) $	$ q >4\\ \text{certain}\ \ \text{ choices}\ \text{ of} \, \delta $	[7,30]
$ 6 $	$ 4 $	$ (x^q, x^{q^3}, x-x^{q^2}, x^{q^4}-\delta x) $	$ \begin{array}{cccc}q \quad \text{odd}\\ \delta^2+\delta =1 \end{array} $	[10,29]
$ 6 $	$ 4 $	$ \begin{array}{cc} (h^{q^2-1}x^q+h^{q-1}x^{q^2}, x^{q^3}, \\ x^q-h^{q-1}x^{q^4}, x^q-h^{q-1}x^{q^5}) \end{array} $	$ \begin{array}{cccc}q \quad \text{odd}\\ h^{q^3+1}=-1 \end{array} $	[3,37]
$ 7 $	$ 3 $	$ (x, x^q, x^{q^3}) $	$ \begin{array}{cc} q \quad \text{odd}, \\ \gcd(s, 7)=1\end{array} $	[8]
$ 7 $	$ 4 $	$ (x, x^{q^{2s}}, x^{q^{3s}}, x^{q^{4s}}) $	$ \begin{array}{cc} q \quad \text{odd}, \\ \gcd(s, n)=1\end{array} $	[8]
$ 8 $	$ 3 $	$ (x, x^q, x^{q^3}) $	$ \begin{array}{cc} q \equiv 1 \pmod{3}, \\ \gcd(s, 8)=1\end{array} $	[8]
$ 8 $	$ 5 $	$ (x, x^{q^{2s}}, x^{q^{3s}}, x^{q^{4s}}, x^{{5s}}) $	$ \begin{array}{cc} q \equiv 1 \pmod{3}, \\ \gcd(s, 8)=1 \end{array} $	[8]
$ 8 $	$ 6 $	$ (x^q, x^{q^2}, x^{q^3}, x^{q^5}, x^{q^6}, x-\delta x^{q^4}) $	$ \begin{array}{cc} q\, \text{odd}, \\ \delta^2=-1\end{array} $	[7]

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