The distance of a stabilizer quantum code is a very important feature since it determines the number of errors that can be detected and corrected. We present three new fast algorithms and implementations for computing the symplectic distance of the associated classical code. Our new algorithms are based on the Brouwer–Zimmermann algorithm. Our experimental study shows that these new implementations are much faster than current state-of-the-art licensed implementations on single-core processors, multicore processors, and shared-memory multiprocessors. In the most computationally-demanding cases, the performance gain in the computational time can be larger than one order of magnitude. The experimental study also shows a good scalability on shared-memory parallel architectures.

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.

The minimum distance of a linear code is a key concept in information theory. Therefore, the time required by its computation is very important to many problems in this area. In this article, we introduce a family of implementations of the Brouwer–Zimmermann algorithm for distributed-memory architectures for computing the minimum distance of a random linear code over 𝔽 2 . Both current commercial and public-domain software only work on either unicore architectures or shared-memory architectures, which are limited in the number of cores/processors employed in the computation. Our implementations focus on distributed-memory architectures, thus being able to employ hundreds or even thousands of cores in the computation of the minimum distance. Our experimental results show that our implementations are much faster, even up to several orders of magnitude, than current implementations widely used nowadays.

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/.

The minimum distance of an error-correcting code is an important concept in information theory. Hence, computing the minimum distance of a code with a minimum computational cost is crucial to many problems in this area. In this article, we present and assess a family of implementations of both the brute-force algorithm and the Brouwer-Zimmermann algorithm for computing the minimum distance of a random linear code over F 2 that are faster than current implementations, both in the commercial and public domain. In addition to the basic sequential implementations, we present parallel and vectorized implementations that produce high performances on modern architectures. The attained performance results show the benefits of the developed optimized algorithms, which obtain remarkable improvements compared with state-of-the-art implementations widely used nowadays.

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.

We study subfield-subcodes of Generalized Toric (GT) codes over Fps. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes. We identify polynomial generators for subfield-subcodes of GT codes which allows us to determine the dimensions and obtain bounds for the minimum distance. We give several examples of binary and ternary subfield-subcodes of GT codes that are the best known codes of a given dimension and length.