On a class of algebraic geometric codes

In the paper, we present a family of algebraic geometric codes over GF(256) that lie above the Gilbert — Varshamov bound together with dual codes. The family of these codes can be used to construct a post-quantum algorithm of the classic McEllice type. The following theorem holds for them: Let $E: y^3=(x^{63}-1)/(x^3-1)$ be a curve over the field $F=\text{GF}(256)$, $P_1,\ldots ,P_{720}$ are arbitrary distinct $F$-rational points of this curve, $P_\infty$ is the point at infinity. Then the algebraic geometric codes $C_r(D,G)$ on the curve defined by the divisors $D=P_1+\ldots +P_{720}$ and $G=rP_\infty$ for all integers $r$, $81\le r\le 197$, are $[720,3r-57,720-3r]_{2^8}$-codes, and their cardinality, as well as the cardinality of their dual $[720,777-3r,3r-114]_{2^8}$-codes, satisfies the Gilbert — Varshamov bound.