Five-Lee-weight linear codes over $ \mathbb{F}_{q}+u\mathbb{F}_{q} $

Pavan Kumar; Noor Mohammad Khan

doi:10.3934/amc.2025002

Article Contents

2025, Volume 19, Issue 5: 1417-1432. Doi: 10.3934/amc.2025002

This issue Previous Article On symbol-pair distance distributions of repeated-root constacyclic codes of length $ 4p^s $ and MDS codes Next Article A digital signature scheme based on the vector space factorization problem and the MPC-in-the-Head paradigm

Five-Lee-weight linear codes over $ \mathbb{F}_{q}+u\mathbb{F}_{q} $

Pavan Kumar^{1, 2, ,} and
Noor Mohammad Khan^1,

1.
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2.
Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru 560012, India

^*Corresponding author: Pavan Kumar
^*Corresponding author: Pavan Kumar

Received: June 2024

Revised: December 2024

Early access: January 14, 2025

Published online: January 14, 2025

The first author was supported by [University Grants Commission, New Delhi, India, under JRF in Science, Humanities & Social Sciences scheme with Grant number 11-04-2016-413564.].

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Abstract

In this study, linear codes having their Lee-weight distributions over the semi-local ring $ \mathbb{F}_{q}+u\mathbb{F}_{q} $ with $ u^{2} = 1 $ are constructed using the defining set and Gauss sums for an odd prime $ q $. Moreover, we derive complete Hamming-weight enumerators for the images of the constructed linear codes under the Gray map. We finally show an application to secret sharing schemes.

Keywords:

Mathematics Subject Classification: Primary: 11T71; Secondary: 94B05.

Citation:

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Table 1. Lee-weight distribution of the code $ \mathcal{C}_{D} $, for $ m\geq3 $ odd

Weight $ w $	Multiplicity $ f $
0	1
$ 2(q-1)q^{2m-3} $	$ (2q-1)q^{2m-2}-2(q-1)q^{m-1}-1 $
$ 2(q-1)(q^{2m-3}-q^{\frac{3m-5}{2}}) $	$ (q-1)(q^{m-1}-q^{\frac{m-1}{2}}) $
$ 2(q-1)(q^{2m-3}+q^{\frac{3m-5}{2}}) $	$ (q-1)(q^{m-1}+q^{\frac{m-1}{2}}) $
$ 2(q-1)(q^{2m-3}-q^{m-2}) $	$ \frac{(q-1)^{2}}{2}(q^{2m-2}+q^{m-1}) $
$ 2(q-1)(q^{2m-3}+q^{m-2}) $	$ \frac{(q-1)^{2}}{2}(q^{2m-2}-q^{m-1}) $

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Table 2. Lee-weight distribution of the code $ \mathcal{C}_{D} $, for $ m\geq2 $ even

Weight w	Multiplicity $ f $
0	1
$ 2(q-1)q^{2m-3}+2G(q-1)^{2}q^{m-3} $	$ 2(q^{m-1}+q^{-1}(q-1)G-1) $
$ 2(q-1)q^{2m-3}+2G(q-1)(2q-1)q^{m-3}+2(q-1)^{2}q^{m-2} $	$ 2(q-1)(q^{m-1}-q^{-1}G) $
$ 2(q-1)q^{2m-3}+4G(q-1)^{2}q^{m-3} $	$ (q^{m-1}+q^{-1}(q-1)G-1)^{2} $
$ 2(q-1)q^{2m-3}+4G(q-1)^{2}q^{m-3}+2(q-1)^{2}q^{m-2} $	$ 2(q-1)(q^{m-1}-q^{-1}G)(q^{m-1}+q^{-1}(q-1)G-1) $
$ 2(q-1)q^{2m-3}+4G(q-1)^{2}q^{m-3}+2(q-1)(q-2)q^{m-2} $	$ (q-1)^{2}(q^{m-1}-q^{-1}G)^{2} $

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