Abstract We study monomial-Cartesian codes (MCCs) which can be regarded as $$(r,\delta )$$ ( r , δ ) -locally recoverable codes (LRCs). These codes come with a natural bound for their minimum distance and we determine those giving rise to $$(r,\delta )$$ ( r , δ ) -optimal LRCs for that distance, which are in fact $$(r,\delta )$$ ( r , δ ) -optimal. A large subfamily of MCCs admits subfield-subcodes with the same parameters of certain optimal MCCs but over smaller supporting fields. This fact allows us to determine infinitely many sets of new $$(r,\delta )$$ ( r , δ ) -optimal LRCs and their parameters.

Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal $$q^2$$ q 2 -ary linear codes. This result can be generalized to $$q^{2 m}$$ q 2 m -ary linear codes, $$m > 1$$ m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de , 2020) and new q-ary ones, with $$q \ne 2$$ q ≠ 2 , improving others in the literature.

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.

We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field F4which are records at http://codetables.de/.

Two new constructions of linear code pairs C2⊂ C1are given for which the codimension and the relative minimum distances M1(C1, C2) and M1(C2⊥, C1⊥) are good. By this, we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes. They can also be used to ensure reliable communication over asymmetric quantum channels. The new constructions result from carefully applying the Feng-Rao bounds to a family of codes defined from multivariate polynomials and Cartesian product point sets.