Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case

We consider linear codes over a finite field $\mathbb{F}_q$, for odd $q$, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a codeword is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upperbounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order $5$, are given.