[Paper Review] A class of three-weight and five-weight linear codes

Fei Li, Qiuyan Wang+1 more|arXiv (Cornell University)|Sep 21, 2015

Coding theory and cryptography2 references3 citations

TL;DR

This paper constructs a class of linear codes over $\mathbb{F}_p\uf8e5$ with three or five weights by defining a code ${\mathcal{C}}_D$ using the trace function over a specific defining set $D = \{x \in \mathbb{F}_q^* : \mathrm{Tr}(x^2 + x) = 0\}$, where $q = p^m$. The key contribution is the complete determination of the weight distributions for these codes under various conditions on $m$ and $p$, and it is shown that for $m \geq 5$, the codes satisfy $w_{\min}/w_{\max} > (p-1)/p$, enabling their use in secret sharing schemes.

ABSTRACT

Recently, linear codes with few weights have been widely studied, since they have applications in data storage systems, communication systems and consumer electronics. In this paper, we present a class of three-weight and five-weight linear codes over Fp, where p is an odd prime and Fp denotes a finite field with p elements. The weight distributions of the linear codes constructed in this paper are also settled. Moreover, the linear codes illustrated in the paper may have applications in secret sharing schemes.

Motivation & Objective

• To construct linear codes with few weights (three or five) over finite fields $\mathbb{F}_p$ for applications in coding theory and cryptography.
• To determine the exact weight distribution of these codes for different cases of $m$ and $p$, particularly when $m$ is even or odd and $p$ divides or does not divide $m$.
• To establish conditions under which the codes can be used in secret sharing schemes by analyzing the ratio $w_{\min}/w_{\max}$.
• To generalize previous constructions of few-weight codes using trace functions and defining sets in finite fields.

Proposed method

• The code ${\mathcal{C}}_D$ is constructed as $\{ (\mathrm{Tr}(x d_1), \dots, \mathrm{Tr}(x d_n)) \mid x \in \mathbb{F}_q \}$, where $D$ is the set of nonzero elements in $\mathbb{F}_q$ satisfying $\mathrm{Tr}(x^2 + x) = 0$.
• The weight distribution is derived using properties of Gauss sums and character sums, particularly involving the quadratic character $\eta$ and the trace function $\mathrm{Tr}$.
• The number of codewords of each weight is computed via Lemmas 14, 15, 17, 18, and 24, which relate the weight to the trace of $x d_i$ and the value of $\mathrm{Tr}(b)$ and $\mathrm{Tr}(b^2)$ for $b \in \mathbb{F}_q$.
• The parameters of the code, including length, dimension, and minimum distance, are determined based on the size of the defining set $D$ and the trace function's image.
• The weight enumerator is computed by solving a system of equations derived from the first two Pless Power Moments, ensuring consistency with the total number of codewords and the sum of weights.
• Theoretical analysis uses the identity $G = -(-1)^{m(p-1)/4} p^{m/2}$ to express the weight multiplicities in terms of $p$, $m$, and Gauss sums.

Experimental results

Research questions

• RQ1 What is the weight distribution of the linear code ${\mathcal{C}}_D$ defined by $D = \{x \in \mathbb{F}_q^* : \mathrm{Tr}(x^2 + x) = 0\}$ over $\mathbb{F}_p$?
• RQ2 How do the parameters and weight distribution of ${\mathcal{C}}_D$ vary depending on whether $m$ is even or odd and whether $p$ divides $m$?
• RQ3 Can the constructed codes with three or five weights be used in secret sharing schemes, and under what conditions?
• RQ4 What is the exact value of the minimum distance and the ratio $w_{\min}/w_{\max}$ for these codes, and does it exceed $(p-1)/p$?
• RQ5 How do Gauss sums and character sums contribute to the exact computation of the weight distribution?

Key findings

• For $m > 2$ even and $p \mid m$, the code ${\mathcal{C}}_D$ is a $[p^{m-1} - 1 + p^{-1}(p-1)G, m]$ code with three weights, where $G = -(-1)^{m(p-1)/4} p^{m/2}$.
• For $m$ even and $p \nmid m$, the code ${\mathcal{C}}_D$ is a $[p^{m-1} - p^{-1}G - 1, m]$ code with five weights, and the weight distribution is fully determined in Table 2.
• When $m$ is odd and $p \mid m$, the code ${\mathcal{C}}_D$ has parameters $[p^{m-1} - 1, m]$ and five distinct nonzero weights, as detailed in Table 3.
• For $m \geq 5$, the ratio $w_{\min}/w_{\max}$ of the code exceeds $(p-1)/p$, satisfying a necessary condition for constructing secret sharing schemes with interesting access structures.
• The code with parameters $(p,m) = (3,6)$ has length 260, dimension 6, minimum distance 162, and weight enumerator $1 + 98x^{162} + 324x^{171} + 306x^{180}$.
• The code with parameters $(p,m) = (3,4)$ has length 29, dimension 4, minimum distance 18, and is optimal according to codetables, with weight enumerator $1 + 44x^{18} + 30x^{21} + 6x^{24}$.