An equivalence relation called isometry is introduced to classify constacyclic codes over a finite field; the polynomial generators of constacyclic codes of length ℓ t p s are characterized, where p is the characteristic of the finite field and ℓ is a prime different from p.

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. MDS codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin ([8], IEEE Trans. Inf. Theory, 2016) constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results in [8] are extended.

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. MDS symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes.

Properties of matrix product codes over finite commutative Frobenius rings are investigated. The minimum distance of matrix product codes constructed with several types of matrices is bounded in different ways. The duals of matrix product codes are also explicitly described in terms of matrix product codes.

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In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. For odd prime power q with q square, the total number of lengths for MDS self-dual codes over F q presented in this paper is much more than those in all the previous results.

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Stereographic projection is one of the most powerful research tools for crystallography in materials science. A new program for full operation of stereographic projections and in‐depth exploration of crystallographic orientation relationships is described. It is specifically designed for materials researchers who are in need of tools for extensive crystallographic analysis. The difference from other popular commercial software for crystallography is that this program provides new options for users to plot and fully control stereographic projections of an arbitrary pole centre for an arbitrary crystal structure and to illustrate composite stereographic projections, which are necessary to explore the orientation relationships between two phases. The program is able to perform a range of essential crystallographic calculations.