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In this work, we study quantum error-correcting codes obtained by using Steane-enlargement. We apply this technique to certain codes defined from Cartesian products previously considered by Galindo et al. (IEEE Trans Inf Theory 64(4):2444–2459, 2018. 10.1109/TIT.2017.2755682). We give bounds on the dimension increase obtained via enlargement, and additionally give an algorithm to compute the true increase. A number of examples of codes are provided, and their parameters are compared to relevant codes in the literature, which shows that the parameters of the enlarged codes are advantageous. Furthermore, comparison with the Gilbert–Varshamov bound for stabilizer quantum codes shows that several of the enlarged codes match or exceed the parameters promised by the bound. [ABSTRACT FROM AUTHOR]

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield subcodes of $J$ -affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield subcodes of $J$ -affine variety codes and provide some new and good LCD codes coming from this construction. [ABSTRACT FROM AUTHOR]

We present a new bound for the minimum distance of a general primary linear code. For affine variety codes defined from generalised $$C_{ab}$$ polynomials the new bound often improves dramatically on the Feng-Rao bound for primary codes (Andersen and Geil, Finite Fields Appl 14(1):92-123, ; Geil et al., Lecture Notes in Computer Science 3857: 295-306, ). The method does not only work for the minimum distance but can be applied to any generalised Hamming weight. [ABSTRACT FROM AUTHOR]

Presents information on a study which estimated the minimum distance and generalized Hamming weights for a class of error-correcting codes using a generalization of Bezout's theorem. Duals of evaluation codes; Estimation of the number of common zeros; Weight hierarchies for the codes; Conclusions.