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 $\mathbb{F}_{q}$. 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 $6$ to $q^{2}+1$, 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 $7\le d+2\leq n\le q+\lfloor 2\sqrt{q}\rfloor+\delta(q)-3$, where $\delta(q)=0$ or $1$. 
Baokun Ding, Gennian Ge, Jun Zhang, Tao Zhang and Yiwei Zhang
cs.IT math.IT