New constructions of MDS symbol-pair codes

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 $${\mathbb {F}}_{q}$$Fq. We show that a linear MDS symbol-pair code over $${\mathbb {F}}_{q}$$Fq with pair-distance 5 exists if and only if the length n ranges from 5 to $$q^2+q+1$$q2+q+1. As for codes with pair-distance 6, length ranging from $$q+2$$q+2 to $$q^{2}$$q2, 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 $$d+2$$d+2 with length n satisfying $$7\le d+2\le n\le q+\lfloor 2\sqrt{q}\rfloor +\delta (q)-3$$7≤d+2≤n≤q+⌊2q⌋+δ(q)-3, where $$\delta (q)=0$$δ(q)=0 or 1.