Chordal Construction of Column-Twisted Reed--Solomon Codes

Overview & Synopsis

A $[n,k]_q$ Generalized Reed--Solomon (GRS) code can be represented, up to monomial equivalence, by a projective system consisting of $n$ points on a degree $(k-1)$ rational normal curve (RNC) $C_{k-1}$ in PG$(k-1,q)$. We use the term \textit{Column-Twisted Reed--Solomon (CTRS)} codes broadly for $[n,k]_q$ codes whose projective system contains at least $(k+2)$ points on a $C_{k-1}$ of PG$(k-1,q)$. By a chord of $C_{k-1}$, we mean a line which is either a tangent, a real secant, or an imaginary secant of $C_{k-1}$. Roth and Lempel constructed non-GRS MDS $[n,k]_q$ codes by considering projective systems consisting of $(n-1)$ points on $C_{k-1}$ together with a point on a tangent line. More recently, Liu et al. obtained non-GRS $[n, k]_q$ MDS codes by considering projective systems consisting of $(n-2)$ points on $C_{k-1}$ together with a pair of points on a real secant. In the case when $q$ is odd, the length of these codes satisfies $k+4 \leq n \leq (q+3)/2$. In this article, we construct non-GRS $[n,k]_q$ MDS codes by considering projective systems consisting of $(n-2)$ points on $C_{k-1}$ together with a pair of Galois conjugate points on an imaginary secant. For odd $q$, these codes have lengths satisfying $k+4 \le n\le (q+5)/2$, thereby improving both the attainable lengths and dimension ranges of previous TRS and CTRS constructions. In particular, our code attains the length $(q+5)/2$ for every $3\le k\le (q-3)/2$. We use a geometric argument to establish the non-GRS property of CTRS codes for the range $3\le k \le n-4$, while the traditional method of comparison with the Schur square of GRS codes proves the non-GRS property only for the narrower range $3\le k \le n/2$. We also determine the structure of the dual of any CTRS code.