Three weight ternary linear codes from non-weakly regular bent functions

Rumi Melih Pelen

doi:10.3934/amc.2022020

Article Contents

2024, Volume 18, Issue 3: 711-737. Doi: 10.3934/amc.2022020

This issue Previous Article Three constructions of Golay complementary array sets Next Article Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

Three weight ternary linear codes from non-weakly regular bent functions

Rumi Melih Pelen^,

Erzurum Technical University, Yakutiye, Erzurum, Turkey

^* Corresponding author: Rumi Melih Pelen
^* Corresponding author: Rumi Melih Pelen

Received: May 2021

Revised: December 2021

Early access: April 2022

Published: June 2024

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Abstract

This paper constructs several classes of three-weight ternary linear codes from non-weakly regular dual-bent functions based on a generic construction method. Instead of the whole space, we use the subspaces $ B_{\pm}(f) $ associated with a ternary non-weakly regular dual-bent function $ f $. Unusually, we use the pre-image sets of the dual function $ f^* $ in $ B_{\pm}(f) $ as the defining sets of the corresponding codes. Since the size of the defining sets of the constructed codes is flexible, it enables us to construct several codes with different parameters for a fixed dimension. We represent the weight distribution of the constructed codes, and we also give several examples.

Keywords:

Mathematics Subject Classification: Primary: 94A05, 94A11, 94B05; Secondary: 94A24, 94A62.

Citation:

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Table 1. The weight distribution of $ \mathcal {C}_{C_{j_0}(f)} $ when $ n $ is even

Hamming weight $ a $	Multiplicity $ E_a $
0	1
$ 23^{r-2} $	$ 3^{2r-n-1}+3^{r-\frac{n}{2}-1} $
$ 2(3^{r-2}+3^{\frac{n}{2}-1}) $	$ 23^{2r-n-1}-23^{r-\frac{n}{2}-1} $
$ 2(3^{r-2}-3^{\frac{n}{2}-2}+3^{\frac{n}{2}-1}) $	$ 3^r-3^{2r-n} $

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Table 2. The weight distribution of $ \mathcal {C}_{C_{j_0+2}(f)} $ when $ n $ is odd

Hamming weight $ a $	Multiplicity $ E_a $
0	1
$ 23^{r-2} $	$ 3^{2r-n-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n-3}{2}}\right) $	$ 3^{r}-23^{2r-n-1}-3^{r-\frac{n+1}{2}} $
$ 2\left(3^{r-2}+23^{\frac{n-3}{2}}\right) $	$ 3^{2r-n-1}+3^{r-\frac{n+1}{2}} $

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Table 3. The weight distribution of $ \mathcal {C}_{D_{j_0+2}(f)} $ when $ n $ is even

Hamming weight $ a $	Multiplicity $ E_a $
0	1
$ 23^{r-2} $	$ 23^{2r-n-1}-3^{r-\frac{n}{2}-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n}{2}-1}\right) $	$ 3^{2r-n-1}+3^{r-\frac{n}{2}-1} $
$ 2\left(3^{r-2}+3^{\frac{n}{2}-2}\right) $	$ 3^r-3^{2r-n} $

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Table 4. The weight distribution of $ \mathcal {C}_{D_{j_0+1}(f)} $ when $ n $ is odd

Hamming weight $ a $	Multiplicity $ E_a $
0	1
$ 23^{r-2} $	$ 3^{2r-n-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n-3}{2}}\right) $	$ 3^{r}-23^{2r-n-1}-3^{r-\frac{n+1}{2}} $
$ 2\left(3^{r-2}+23^{\frac{n-3}{2}}\right) $	$ 3^{2r-n-1}+3^{r-\frac{n+1}{2}} $

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