New constructions of MDS symbol-pair codes

Abstract

Motivated by the application of high-density data storage technologies, symbol-pair codes are proposed to protect against pair-errors in symbol-pair channels, whose outputs are overlapping pairs of symbols. The research of symbol-pair codes with the largest minimum pair-distance is interesting since such codes have the best possible error-correcting capability. A symbol-pair code attaining the maximal minimum pair-distance is called a maximum distance separable (MDS) symbol-pair code. In this paper, we focus on constructing linear MDS symbol-pair codes over the finite field ${\mathbb {F}}_{q}$. We show that a linear MDS symbol-pair code over ${\mathbb {F}}_{q}$ with pair-distance 5 exists if and only if the length n ranges from 5 to $q^2+q+1$. As for codes with pair-distance 6, length ranging from $q+2$ to $q^{2}$, we construct linear MDS symbol-pair codes by using a configuration called ovoid in projective geometry. With the help of elliptic curves, we present a construction of linear MDS symbol-pair codes for any pair-distance $d+2$ with length n satisfying $7\le d+2\le n\le q+\lfloor 2\sqrt{q}\rfloor +\delta (q)-3$, where $\delta (q)=0$ or 1.