Let p be an odd prime, s and m be positive integers. Cyclic codes of length 2psover Fpmare the ideals ((x-1)i(x+1)j), where 0 ≤ i, j ≤ ps, of the principal ideal ring Fpm[x]/(x2ps- 1). Using this structure, the symbol-pair distances of all cyclic codes of length 2psover Fpmare completely determined. In addition, we establish all MDS symbol-pair cyclic codes of length 2psover Fpm. Some MDS symbol-pair cyclic codes are better than all the known ones. Among others, we discuss possible applications to construct quantum MDS symbol-pair codes.

Let p be a prime, and λ be a nonzero element of the finite field Fpm. The λ-constacyclic codes of length psover Fpmare linearly ordered under set-theoretic inclusion, i.e., they are the ideals 〈(x - λ0)i〉, 0 ≤ i ≤ psof the chain ring [(Fpm[x])/((xps- λ))]. This structure is used to establish the symbol-pair distances of all such λ-constacyclic codes. Among others, all maximum distance separable symbol-pair constacyclic codes of length ps are obtained.