Improvement on Minimum Distance of Symbol-Pair Codes

Abstract

Symbol-pair codes were first introduced by Cassuto and Blaum (2010). The minimum pair distance of a code is a criterion that characterises the error correcting capability of the code with respect to pair errors. The codes that achieve the optimal minimum pair distance (for given codeword length, code book size and alphabet) are called Maximum Distance Separable (MDS) symbol-pair codes. A way to study the minimum pair distance of a code is through its connection to the minimum Hamming distance of the code. For certain structured codes, these two types of distances can be very different. Yaakobi et al. (2016) showed that for a binary cyclic code, the minimum pair distance is almost three halves of its minimum Hamming distance. We extend this connection to q-ary (q is a prime power) constacyclic codes. The extension involves non-trivial usage of the double counting technique in combinatorics. Such a connection naturally yields a constructive lower bound on the minimum pair distance of q-ary symbol-pair codes. For some choices of the code parameters, this lower bound matches the Singleton type upper bound, yielding q-ary MDS symbol-pair codes.

Appendix

For better understanding,we give detailed proof of Lemma 1.

Proof

Let $\varvec{x}=(x_0,x_1,\cdots ,x_{n-1})\in \varSigma ^n.$ Our goal is to calculate $w_p(\varvec{x}),$ namely,

$$w_p(\varvec{x})=wt\{(x_0,x_1),(x_1,x_2),\cdots ,(x_{n-1},x_0)\}.$$

Now we let

$S_0=\{i:(x_i,x_{i+1})\ne (0,0)\ \text {and}\ x_i=1\},$

$S_1=\{i:(x_i,x_{i+1})= (0,1)\}.$

In the cases above, $S_0$ contains all the pairs (1, 0) and (1, 1). $S_1$ contains all the pairs (0, 1). Hence, $|S_0|=\omega _H(\varvec{x})$, $S_0\cap S_1=\emptyset$, and $\omega _p(\varvec{x})=|S_0|+|S_1|$. For all $0\le i\le {n-1}$, $i\in S_1$ if and only if $x_{i+1}=1$ and $x_i=0$. Thus, $x_i+x_{i+1}=1$ or $x_{i+1}^{'}=1$, where $x_i=0$. Hence, we get

$$\begin{aligned} |S_1|=|\{i: x_{i+1}=1\ \text {and} \ x_i=0\}|. \end{aligned}$$

Note that for any $\varvec{x}\in \varSigma ^n$,

$$\begin{aligned} |\{i: x_{i+1}=1\ \text {and}\ x_i=0\}|=|\{i:x_{i+1}=0\ \text {and}\ x_i=1\}|, \end{aligned}$$

and the sum of the cardinality of the two sets is $\omega _H(\varvec{x}^{'})$. Hence, $S_1=\frac{\omega _H(\varvec{x}^{'})}{2}$ and

$$\begin{aligned} \omega _p(\varvec{x}) =|S_0|+|S_1|=\omega _H(\varvec{x})+\frac{\omega _H(\varvec{x}^{'})}{2}. \end{aligned}$$