🤖 AI Summary
This study investigates the weight distribution of generalized doubly extended Reed–Solomon codes with minimum distance \(d \geq 5\) over cosets of weight 2. By leveraging finite field theory, orbit analysis under group actions, and algebraic combinatorial techniques, the authors establish for the first time that Case S holds—implying the code is 2-regular—whenever \(q-1\) and \(d-2\) are coprime. They further introduce two equivalent new combinatorial problems, denoted \(A_{q,\mu}^\times\) and \(A_{R,\mu}^+\), which provide a viable pathway toward resolving Case NS. The proposed general framework successfully computes complete weight distributions for multiple \((q,d)\) parameter pairs, thereby solving the long-standing problem of determining sufficient conditions for Case S.
📝 Abstract
We consider the weight distributions of the cosets of weight 2 of the generalized $[q+1,q+2-d,d]_q$ doubly extended Reed-Solomon codes (GDRS) of minimum distance $d\ge5$, over the finite field $\mathbb{F}_q$ with $q$ elements. For a GDRS code, we say that Case S occurs if the weight distribution for all cosets of weight 2 is the same or otherwise, Case NS occurs. For Case S, the weight distribution is known; however, any sufficient condition for the occurrence of Case S remained an open problem. We prove that if $q-1$ and $d-2$ are coprime then Case S holds, i.e. the problem is solved. Furthermore, we note that in Case S, the GDRS code is 2-regular. Also, we introduce two new open equivalent combinatorial problems for finite fields $\mathbb{F}_q$ (Problem $A_{q,μ}^\times$) and for rings $\mathbb{Z}_\mathfrak{R}$ of integers modulo $\mathfrak{R}$ (Problem $A_{\mathfrak{R},μ}^+$), where $μ$ is a parameter. In particular, Problem $A_{\mathfrak{R},μ}^+$ is as follows: for each element $λ$ of $\mathbb{Z}_\mathfrak{R}$, determine the number of all possible $μ$-tuples $\{λ_1,λ_2,\ldots,λ_μ\}$, each of which consists of $μ$ distinct elements $λ_j$ of $\mathbb{Z}_\mathfrak{R}$ such that their sum in $\mathbb{Z}_\mathfrak{R}$ is equal to $λ$. Open Problems $A_{q,μ}^\times$ and $A_{\mathfrak{R},μ}^+$ are interesting in their own right and, moreover, we proved that their solutions allow us to obtain the weight distributions for Case NS, taking $μ=d-2$ and $\mathfrak{R}=q-1$. To solve Problem $A_{\mathfrak{R},μ}^+$, we found a universal method, connected with the values of $\mathfrak{R}$ and $μ$, using orbits of elements in $\mathbb{Z}_\mathfrak{R}$ and then we solved the problem for many pairs $\mathfrak{R},μ$, obtaining the needed weight distributions for the corresponding pairs $q=\mathfrak{R}+1,d=μ+2$.