New Bounds on the Code Size of Symbol-Pair Codes

Classical error-correcting codes under the Hamming metric are used to correct substitution and erasure errors. Motivated by the limitations of the reading process in high density data storage systems, a new class of codes called symbol-pair (metric) codes was designed to protect against pair errors in symbol-pair read channels. For a given alphabet of size $q$ and given values of $n$ and $d$ with $1\leq d\leq n$ , let $A_{p}(n,d,q)$ denote the largest possible code size for which there exists a $q$ -ary code of length $n$ with minimum pair-distance at least $d$ . In this paper, new upper and lower bounds on $A_{p}(n,d,q)$ are presented. Several examples are included to illustrate our main results; some examples are optimal in the sense that they meet the corresponding bounds, and the rest examples are meant to show that our bounds may perform better than some of the previously known ones in certain cases. In addition, we show that any symbol-pair code over $\mathbb {F}_{q}$ can be viewed as a Hamming metric code over $\mathbb {F}_{q^{2}}$ with the same parameters. Consequently, the theory of classical codes over $\mathbb {F}_{q^{2}}$ can be used directly to symbol-pair codes; in particular, by virtue of this result, some known results can be reobtained immediately.