New ternary self-orthogonal codes and related LCD codes from weakly regular plateaued functions

Dengcheng Xie; Shixin Zhu; Yang Li

doi:10.3934/amc.2025004

Article Contents

2025, Volume 19, Issue 5: 1460-1487. Doi: 10.3934/amc.2025004

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New ternary self-orthogonal codes and related LCD codes from weakly regular plateaued functions

School of Mathematics, Hefei University of Technology, Hefei 230601, Anhui, China

^*Corresponding author: Shixin Zhu
^*Corresponding author: Shixin Zhu

Received: July 2024

Revised: January 2025

Early access: January 24, 2025

Published online: January 24, 2025

This research is supported by the National Natural Science Foundation of China (Grant Nos. U21A20428 and 12171134) and Anhui Provincial Natural Science Foundation (Grant No. 2408085MA014) and the Fundamental Research Funds for the Central Universities (Grant No. JZ2024HGTB0203) and the Natural Science Research Key Project of Anhui Educational Committee (Grant No. 2024AH050303).

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Abstract

A linear code is said to be self-orthogonal if it is contained in its dual. Self-orthogonal codes are of interest because of their important applications, such as for constructing linear complementary dual (LCD) codes and quantum codes. In this paper, we construct several new families of ternary self-orthogonal codes by employing weakly regular plateaued functions. Their parameters and weight distributions are completely determined. Then we apply these self-orthogonal codes to construct several new families of ternary LCD codes. As a consequence, we obtain many (almost) optimal ternary self-orthogonal codes and LCD codes.

Keywords:

Mathematics Subject Classification: Primary: 94B05; Secondary: 12E10.

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Table 1. The weight distribution of $ \overline{\mathcal{C}_{D_{fg}(0)}} $ when $ s+k_f+k_g $ is even in Theorem $ \rm 4.2 $

Weight $ i $	Multiplicity $ A_i $
$ 0 $	$ 1 $
$ 2\cdot3^{s-2}+2\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-2}{2}} $	$ 3^{s-k_f-k_g-1}-(-1)^{s+1}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-2}{2}} $
$ 2\cdot3^{s-2}+\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-2}{2}} $	$ 4\cdot3^{s-k_f-k_g-1}+2(-1)^{s+1}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-2}{2}}-2 $
$ 2\cdot 3^{s-2} $	$ 4 \cdot 3^{s-k_f-k_g-1}-(-1)^{s+1}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-2}{2}}-1 $
$ 2\cdot3^{s-2}-2\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-4}{2}} $	$ 3^{s+1}-3^{s-k_f-k_g+1} $
$ 3^{s-1}+\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-2}{2}} $	$ 2 $

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Table 2. The weight distribution of $ \overline{\mathcal{C}_{D_{fg}(0)}} $ when $ s+k_f+k_g $ is odd in Theorem 4.4

Weight $ i $	Multiplicity $ A_i $
$ 0 $	$ 1 $
$ 2\cdot3^{s-2}+2\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-3}{2}} $	$ 3^{s+1}-2\cdot3^{s-k_f-k_g}+\eta_0(\lambda)(-1)^{s}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g+1}{2}} $
$ 2\cdot3^{s-2} $	$ 3^{s-k_f-k_g-1}-1 $
$ 2\cdot 3^{s-2}+4\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-3}{2}} $	$ 3^{s-k_f-k_g-1}+\eta_0(\lambda)(-1)^{s}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-1}{2}} $
$ 2\cdot 3^{s-2}-\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-1}{2}} $	$ 2 \cdot 3^{s-k_f-k_g-1}-2 $
$ 2\cdot 3^{s-2}+\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-3}{2}} $	$ 2\cdot 3^{s-k_f-k_g-1}+2\eta_0(\lambda)(-1)^{s}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-1}{2}} $
$ 3^{s-1}-\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-1}{2}} $	$ 2 $

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Table 3. The weight distribution of $ \overline{\mathcal{C}_{D_{fg}(0)}} $ when $ s+k_f+k_g $ is even in Theorem 4.8

Weight $ i $	Multiplicity $ A_i $
$ 0 $	$ 1 $
$ 2\cdot3^{s-2}-4\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-4}{2}} $	$ 3^{s-k_f-k_g-1}-(-1)^{s+1}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-2}{2}} $
$ 2\cdot3^{s-2}-\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-4}{2}} $	$ 4\cdot3^{s-k_f-k_g-1}+2(-1)^{s+1}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-2}{2}} $
$ 2\cdot 3^{s-2}+2\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-4}{2}} $	$ 4 \cdot 3^{s-k_f-k_g-1}-(-1)^{s+1}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-2}{2}} $
$ 2\cdot3^{s-2} $	$ 3^{s+1}-3^{s-k_f-k_g+1}-3 $
$ 3^{s-1} $	$ 2 $

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Table 4. The weight distribution of $ \overline{\mathcal{C}_{D_{fg}(0)}} $ when $ s+k_f+k_g $ is odd in Theorem 4.9

Weight $ i $	Multiplicity $ A_i $
$ 0 $	$ 1 $
$ 2\cdot3^{s-2} $	$ 3^{s+1}-2\cdot3^{s-k_f-k_g}+\eta_0(\lambda)(-1)^{s}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g+1}{2}}-3 $
$ 2\cdot3^{s-2}-2\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-3}{2}} $	$ 3^{s-k_f-k_g-1} $
$ 2\cdot 3^{s-2}+2\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-3}{2}} $	$ 3^{s-k_f-k_g-1}+\eta_0(\lambda)(-1)^{s}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-1}{2}} $
$ 2\cdot 3^{s-2}+\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-3}{2}} $	$ 2 \cdot 3^{s-k_f-k_g-1} $
$ 2\cdot 3^{s-2}-\eta_0(\lambda)\varepsilon_f\varepsilon_g{(-3)}^{\frac{s+k_f+k_g-3}{2}} $	$ 2\cdot 3^{s-k_f-k_g-1}+2 \eta_0(\lambda)(-1)^{s}\varepsilon_f\varepsilon_g{(-3)}^{\frac{s-k_f-k_g-1}{2}} $
$ 3^{s-1} $	$ 2 $

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Table 5. The weight distribution of $ \overline{\mathcal{C}_{D_{g}(0)}} $ when $ m+k_g $ is even

Weight $ i $	Multiplicity $ A_i $
$ 0 $	$ 1 $
$ 2\cdot3^{s-2}-2\cdot3^{n-2}\varepsilon_g{(-3)}^{\frac{m+k_g}{2}} $	$ 2\cdot3^{m-k_g} $
$ 2\cdot3^{s-2}+3^{n-2}\varepsilon_g{(-3)}^{\frac{m+k_g}{2}} $	$ 4\cdot3^{m-k_g} $
$ 2\cdot3^{s-2} $	$ 3^{s+1}-2\cdot3^{m-k_g+1}-3 $
$ 3^{s-1} $	$ 2 $

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Table 6. The weight distribution of $ \overline{\mathcal{C}_{D_{g}(0)}} $ when $ m+k_g $ is odd

Weight $ i $	Multiplicity $ A_i $
$ 0 $	$ 1 $
$ 2\cdot3^{s-2}-3^{n-2}\varepsilon_g{(-3)}^{\frac{m+k_g+1}{2}} $	$ 2\cdot3^{m-k_g} $
$ 2\cdot3^{s-2}+3^{n-2}\varepsilon_g{(-3)}^{\frac{m+k_g+1}{2}} $	$ 2\cdot3^{m-k_g} $
$ 2\cdot3^{s-2} $	$ 3^{s+1}-4\cdot3^{m-k_g}-3 $
$ 3^{s-1} $	$ 2 $

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