Symbol-pair distance of some repeated-root constacyclic codes of length ps over the Galois ring GR(pa, m)

Let a and m be positive integers and lambda be any unit in GR(p(a), m) of the form lambda = (sigma(0) + p sigma(1) + p(2)z), where sigma(0), sigma(1) is an element of T(p, m) \ {0} and z is an element of GR(p(a), m). The symbol-pair distance of all such lambda-constacyclic codes over GR(p(a), m) of length ps are determined. As an application, we identify all maximum distance separable (MDS) lambda-constacyclic codes of length p(s) over GR(p(a), m) with respect to the symbol-pair distance. We give numerous examples of newly constructed MDS symbol-pair codes, i.e., new optimal symbol-pair codes with respect to the Singleton bound.