[Paper Review] Two-weight and three-weight codes from trace codes over $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p$

Yan Liu, Minjia Shi+1 more|arXiv (Cornell University)|Dec 1, 2016

Coding theory and cryptography8 references3 citations

TL;DR

This paper constructs an infinite family of two-weight and three-weight codes over the non-chain ring $\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p$ using trace codes and Gauss sums. By applying a linear Gray map, it obtains abelian $p$-ary codes with few weights, and proves that the two-weight codes are optimal via the Griesmer bound. The dual Lee distance is shown to be 2, enabling a dictatorial secret sharing scheme.

ABSTRACT

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p,$ where $u^2=0,v^2=0,uv=vu.$ These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over $\mathbb{F}_p$. In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given.

Motivation & Objective

• To construct new families of linear codes with few weights over a non-chain ring $\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p$.
• To determine the Lee weight distribution of trace codes over this ring using Gauss sums.
• To establish the optimality of the resulting two-weight codes over $\mathbb{F}_p$ via the Griesmer bound.
• To analyze the dual Lee distance and apply the results to secret sharing schemes.
• To generalize prior constructions by extending from cyclic to abelian codes over a more complex ring structure.

Proposed method

• Define trace codes $C(m,p)$ over the ring $\mathcal{R} = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m} + v\mathbb{F}_{p^m} + uv\mathbb{F}_{p^m}$ using the evaluation map $Ev(a) = (\mathrm{Tr}(ax))_{x \in L}$.
• Use Gauss sums to compute the Lee weight distribution of codewords based on the square or non-square status of $\alpha \in \mathbb{F}_{p^m}^*$.
• Apply a linear Gray map $\phi$ to map codes over the ring to $p$-ary codes, preserving the abelian structure.
• Establish the dual Lee distance $d^\prime = 2$ using the sphere-packing bound and unit properties of the ring.
• Leverage the criterion $\omega_0 / \omega_\infty > (p-1)/p$ to prove all nonzero codewords are minimal in certain cases.
• Construct a secret sharing scheme based on the minimal codewords of the image code $\phi(C(m,p))$.

Experimental results

Research questions

• RQ1 Can trace codes over the ring $\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p$ yield codes with few weights?
• RQ2 What is the exact Lee weight distribution of such trace codes, and how does it depend on $m$ and $p$?
• RQ3 Are the resulting $p$-ary codes via Gray map optimal in terms of the Griesmer bound?
• RQ4 What is the dual Lee distance of the constructed codes, and what does it imply for secret sharing?
• RQ5 Can the minimal codewords of the code lead to a dictatorial or democratic secret sharing scheme?

Key findings

• The Lee weight of codewords in $C(m,p)$ depends on the quadratic character of $\alpha$ when $a = \alpha uv$, with weights $2(p-1)(p^{4m-1} \pm \epsilon(p)p^{(7m-2)/2})$ for $m$ singly-even and $p \equiv 1 \pmod{4}$.
• For $m$ odd and $p \equiv 3 \pmod{4}$, the Lee weight of $a = \alpha uv$ is $2(p^{4m} - p^{4m-1})$, while other codewords have weight $2(p-1)(p^{4m-1} - p^{3m-1})$.
• The two-weight codes over $\mathbb{F}_p$ obtained via the Gray map are optimal, as confirmed by the Griesmer bound.
• The dual Lee distance $d^\prime$ of $C(m,p)$ is exactly 2 for all $m > 1$, implying the existence of codewords of weight 2 and no weight 1 codewords in the dual.
• All nonzero codewords of the $p$-ary image code are minimal when $m > 2$ and $m$ is even, or when $m$ is odd and $p \equiv 3 \pmod{4}$, due to the ratio of minimum to maximum weight exceeding $(p-1)/p$.
• A secret sharing scheme constructed from $\phi(C(m,p))$ is dictatorial, as $d^\prime = 2$ implies the presence of 'dictators' who belong to every coalition.