The ring R = Fpm + uFpm has precisely pm(pm–1) units, which are of the forms γ and α+uβ, where 0 ≠ α,β,γ ∈ Fpm. Using generator polynomial structures of constacyclic codes of length ps over R, the Hamming and symbol-pair distance distributions of all such codes are completely determined. As examples, we provide some good codes with better parameters than the known ones.

Symbol-pair codes are used to protect against symbol-pair errors in high density data storage systems. One of the most important tasks in symbol-pair coding theory is to design MDS codes. MDS symbol-pair codes are optimal in the sense that such codes attain the Singleton bound. In this paper, a new class of MDS symbol-pair codes with code-length $5p$ and optimal pair distance of 7 is established. It is shown that for any prime $p \equiv 1 \pmod 5$ , we can always construct four $p$ -ary MDS symbol-pair cyclic codes of length $5p$ of largest possible pair distance 7. We also completely determined all MDS symbol-pair and MDS $b$ -symbol codes of length $p^{s}$ and $2p^{s}$ over $\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$ by filling in some missing cases, and rectifying some gaps in Type 3 codes of recent papers. As an applications of our results, we use MAGMA to provide many examples of new MDS codes over $\mathbb F_{p^{m}}$ and $\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$ .

Let p be an odd prime, and Fpm is the finite field of pm elements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2ps over Fpm are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2ps over finite fields using the Calderbank- Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2ps over Fpm. To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose–Chaudhuri–Hocquenghem (briefly, BCH) codes.

Let p ≠ 3 be any prime. The structures of all λ-constacyclic codes of length 3ps over the finite commutative chain ring Fpm + uFpm(u2 = 0) are established in the term of their generator polynomials. As an application, Hamming and homogeneous distance of a class of such codes and RT distances of all are given. Among such λ-constacyclic codes, the unique maximum-distance-separable (briefly, MDS) code with respect to the RT distance is obtained. Moreover, when λ is not a cube in Fpm, the necessary and sufficient condition for the λ-constacyclic code of length 3ps over Fpm + uFpm(u2 = 0) be an MDS constacyclic code with respect to Hamming distance is provided.