Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of q-ary AQCs is extended by removing the F_q-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure q-ary CSS-like codes for q ∈ {2,3,4,5,7,8,9} up to reasonable lengths are found. In many instances, removing the F_q-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.

The weights in maximum distance separable (MDS) codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n - k + 1 to n. The proof uses the covering radius of the dual code.

Locally repairable codes have become a key instrument in large-scale distributed storage systems. This paper focuses on the construction of locally repairable codes with $(r,\delta)$ -locality that achieve equality in the Singleton-type bound. We use matrix-product codes to propose two constructions of $q$ -ary optimal $(r,\delta)$ locally repairable codes of lengths up to $q^{2}+q$ . The ingredients in the matrix-product codes are linear maximum distance separable codes. We give another construction of optimal $(r,\delta)$ locally repairable codes by using optimal locally repairable codes as ingredients in the matrix-product approach. The codes in this third construction have unbounded lengths not divisible by $(r+\delta -1)$ . The three constructions of optimal $(r,\delta)$ locally repairable codes constructed here are new. Previously constructed codes in the literature have not covered the same sets of parameters. Our construction proposals are flexible since one can easily vary $r$ and $\delta $ to come up with particular parameters that can suit numerous scenarios.

A recent study by one of the authors has demonstrated the importance of profile vectors in DNA-based data storage. We provide exact values and lower bounds on the number of profile vectors for finite values of alphabet size q, read length 1, and word length n. Consequently, we demonstrate that for q ≥ 2 and n ≤ q ^1/2-1 , the number of profile vectors is at least q ^κn with κ very close to 1. In addition to enumeration results, we provide a set of efficient encoding and decoding algorithms for certain families of profile vectors.