Weight distributions of cosets of weight 2 of the generalized doubly extended Reed-Solomon codes
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 $A_{q,\mu}^\times$) and for rings $\mathbb{Z}_\mathfrak{R}$ of integers modulo $\mathfrak{R}$ (Problem $A_{\mathfrak{R},\mu}^+$), where $\mu$ is a parameter. In particular, Problem $A_{\mathfrak{R},\mu}^+$ is as follows: for each element $\lambda$ of $\mathbb{Z}_\mathfrak{R}$, determine the number of all possible $\mu$-tuples $\{\lambda_1,\lambda_2,\ldots,\lambda_{\mu}\}$, each of which consists of $\mu$ distinct elements $\lambda_j$ of $\mathbb{Z}_\mathfrak{R}$ such that their sum in $\mathbb{Z}_\mathfrak{R}$ is equal to $\lambda$. Open Problems $A_{q,\mu}^\times$ and $A_{\mathfrak{R},\mu}^+$ are interesting in their own right and, moreover, we proved that their solutions allow us to obtain the weight distributions for Case NS, taking $\mu=d-2$ and $\mathfrak{R}=q-1$. To solve Problem $A_{\mathfrak{R},\mu}^+$, we found a universal method, connected with the values of $\mathfrak{R}$ and $\mu$, using orbits of elements in $\mathbb{Z}_\mathfrak{R}$ and then we solved the problem for many pairs $\mathfrak{R},\mu$, obtaining the needed weight distributions for the corresponding pairs $q=\mathfrak{R}+1,d=\mu+2$.