The symbol-pair simplex codes were introduced by the authors recently, and these codes play a similar role as simplex codes with respect to the Hamming metric. Among other things, the concatenations of the symbol-pair simplex codes are symbol-pair constant-weight codes which are a family of codes achieving the Plotkin-type upper bound of the generalized symbol-pair weight, and thus provide optimal security in the data transmission with the symbol-pair metric in the wire-tap channel with the coset coding scheme. Motivated by the mentioned applications, we present new constructions of symbol-pair simplex codes over any finite field.

We present the Plotkin-type bound on the generalized symbol-pair weight and show that all the symbol-pair equiweight codes achieve the Plotkin-type bound. Some new judging criteria are given for symbol-pair equiweight codes, and based on these judging criteria, we show the existence of a class of symbol-pair equiweight codes by design and graph theory. Also, we explicitly construct the symbol-pair version simplex code by the action of a group on a set. Furthermore, we construct several classes of inequiweight codes achieving the Plotkin-type bound.