Linear codes associated to symmetric determinantal varieties; General case
Peter Beelen, Trygve Johnsen, Prasant Singh
TL;DR
The paper studies linear codes arising from symmetric determinantal varieties by evaluating all linear homogeneous polynomials on the affine cone over the set of symmetric $m\times m$ matrices with rank at most $t$. It leverages the $Q$-numbers of the association scheme of symmetric matrices and generalized Krawtchouk polynomials to obtain the full weight distribution and express all possible codeword weights $W_k^{\tau}(t,m)$ and $W_k(t,m)$ in terms of $\mu_r(m)$ and the functions $F_r^{(m-1)}(\cdot)$. The main results determine the minimum distance: for even $t$, $d=C_{symm}(2t,m)=W_1(2t,m)=W_2(2t,m)$, and for odd $t$, $d=C_{symm}(2t+1,m)=W_2^{1}(2t+1,m)$, with the weights independent of the type $\tau$ in these cases. These findings confirm a prior conjecture and provide a field-independent proof, while tying the code’s parameters to geometric hyperplane sections of the symmetric determinantal variety. The work yields a compact weight distribution and highlights the interplay between coding-theoretic parameters and the geometry of determinantal varieties.
Abstract
The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank, which is, at most, a given even number. Furthermore, a conjecture for the minimum distance of codes from symmetric matrices with ranks bounded by an odd number was given. In this article, we continue the study of codes from symmetric matrices of bounded rank. A connection between the weights of the codewords of this code and Q-numbers of the association scheme of symmetric matrices is established. Consequently, we get a concrete formula for the weight distribution of these codes. Finally, we determine the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank at most a given number, both when this number is odd and when it is even.