Extension of MDS Codes over Finite Fields

Overview & Synopsis

In this paper, we investigate extending of $[n,k,n-k+1]$ non-Reed--Solomon(non-RS) MDS codes to $[n+1,k,n-k+2]$ non-RS MDS codes. We develop a practical iterative algorithm for constructing longer MDS codes while preserving their non-RS structure. The proposed algorithm is inspired by the meet-in-the-middle algorithm in cryptanalysis. On average, it requires at most $O(kq^k)$ multiplications, compared with $O\left(\binom{n}{k-1} q^{k-1} k^3\right)$ multiplications for generic exhaustive search. Applying the proposed algorithm to several known families of MDS codes, including elliptic-curve, twisted Reed--Solomon, column-twisted Reed--Solomon, cyclic, and missing-monomial MDS codes over fields of order at most $32$, we obtain new MDS codes of increased lengths that remain inequivalent to Reed--Solomon codes. In particular, we construct $[9,4,6]$ codes over $\F_9$ and $\F_{13}$, each with Schur-square dimension $9$. Over $\F_{17}$, we obtain two inequivalent $[10,4,7]$ codes with Schur-square dimensions $9$ and $10$, respectively. The latter is the dual of an extended $[10,6,5]$ code, demonstrating that the method also applies when $k>n/2$. For these dimension-four codes, the Schur-square dimensions $9$ and $10$ exceed both that of RS code (Schur-square dimension $7$) and those of specific elliptic-curve and column-twisted Reed--Solomon families (Schur-square dimension $8$). Some of these parameters of non-RS MDS codes have not been reported before.