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Linear Codes Constructed From Two Weakly Regular Plateaued Functions with Index (p-1)/2

Shudi Yang, Tonghui Zhang, Zheng-An Yao

TL;DR

This work constructs linear codes over $\mathbb{F}_p$ from a defining set $D_{f,g}=\{(x,y)\in \mathbb{F}_q^2\setminus\{(0,0)\}: f(x)+g(y)=0\}$ using two weakly regular plateaued functions with index $\frac{p-1}{2}$ under $p\equiv 1\pmod 4$. Weight distributions are determined through exponential sums and the Walsh transform, yielding many few-weight and minimal codes, with punctured versions that are projective and sometimes optimal for secret sharing and association schemes. The results cover multiple cases for the indices $l_f\in\{2,(p-1)/2,p-1\}$ and $l_g=(p-1)/2$, providing explicit formulas for $N_0$ and the full weight enumerators, and they demonstrate the practical relevance by identifying minimality conditions and optimal punctured codes. Overall, the paper extends prior work by incorporating the $(p-1)/2$ index and offers a comprehensive set of weight-distribution results and constructions with potential applications in cryptography and combinatorial design.

Abstract

Linear codes are the most important family of codes in cryptography and coding theory. Some codes have only a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting $ p\equiv 1 \pmod 4 $, we construct an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index $ (p-1)/2 $. Their weight distributions are completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal.

Linear Codes Constructed From Two Weakly Regular Plateaued Functions with Index (p-1)/2

TL;DR

This work constructs linear codes over from a defining set using two weakly regular plateaued functions with index under . Weight distributions are determined through exponential sums and the Walsh transform, yielding many few-weight and minimal codes, with punctured versions that are projective and sometimes optimal for secret sharing and association schemes. The results cover multiple cases for the indices and , providing explicit formulas for and the full weight enumerators, and they demonstrate the practical relevance by identifying minimality conditions and optimal punctured codes. Overall, the paper extends prior work by incorporating the index and offers a comprehensive set of weight-distribution results and constructions with potential applications in cryptography and combinatorial design.

Abstract

Linear codes are the most important family of codes in cryptography and coding theory. Some codes have only a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting , we construct an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index . Their weight distributions are completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal.
Paper Structure (12 sections, 23 theorems, 81 equations, 16 tables)

This paper contains 12 sections, 23 theorems, 81 equations, 16 tables.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Cyclotomic classes and cyclotomic fields
  4. Exponential sums
  5. Weakly regular plateaued functions
  6. Auxiliary results
  7. Main results
  8. The calculation of $N_0$
  9. Weight distributions of $C_{D_{f,g}}$ from two weakly regular plateaued functions
  10. The punctured code
  11. Minimality of the codes and their applications
  12. Conclusion

Key Result

Lemma 1

(ire2013) The following assertions hold for $K=\mathbb{Q}(\zeta_p)$. $(1)$ The ring of integers in $K$ is $\mathbb{Z}[\zeta_p]$, and $\{ \zeta_p^i:1\leqslant i \leqslant p-1\}$ is an integer basis of $\mathbb{Z}[\zeta_p]$. $(2)$ The field extension $K/\mathbb{Q}$ is Galois of degree $p-1$, and the G

Theorems & Definitions (38)

  • Lemma 1
  • Lemma 2: Theorem 5.51, lidl1997
  • Lemma 3: Theorem 5.33, lidl1997
  • Lemma 4
  • Definition 1
  • Lemma 5
  • Remark 1
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 28 more