Anna-Lena Horlemann

We establish dihedral quantum codes of short block length, a class of Calderbank–Shor–Steane (CSS) codes obtained by the lifted product construction. We present the code construction and give a formula for the code dimension, depending on the two classical codes that the CSS code is based on. We also give a lower bound on the code distance and construct an example of short dihedral quantum codes.

In this paper, we introduce a new variant of the syndrome decoding problem (SDP), called restricted SDP (R-SDP), in which the entries of the solution vector live in a fixed subset of the underlying finite field. We prove the NP-completeness of R-SDP via a reduction from SDP and show how this new problem can be employed in identification schemes. We revisit some concepts of classical coding theory in light of this new setting, provide several bounds such as the Gilbert-Varshamov, the Singleton, and the Plotkin bound, and study the behavior of random codes. Resulting from this initial work, several proposals have arisen; in particular, the code-based digital signature scheme named CROSS, which is currently competing within NIST's additional call for post-quantum digital signatures, is based on R-SDP.

We introduce a new weight and corresponding metric over finite extension fields for asymmetric error correction. The weight distinguishes between elements from the base field and the ones outside of it, which is motivated by asymmetric quantum codes. We set up the theoretic framework for this weight and metric, including upper and lower bounds, asymptotic behavior of random codes, and we show the existence of an optimal family of codes achieving the Singleton-type upper bound.

We study the distinguishability of linearized Reed-Solomon (LRS) codes by defining and analyzing analogs of the square-code and the Overbeck distinguisher for classical Reed-Solomon and Gabidulin codes, respectively. Our main results show that the square-code distinguisher works for generalized linearized Reed-Solomon (GLRS) codes defined with the trivial automorphism, whereas the Overbeck-type distinguisher can handle LRS codes in the general setting. We further show how to recover defining code parameters from any generator matrix of such codes in the zero-derivation case. For other choices of automorphisms and derivations simulations indicate that these distinguishers and recovery algorithms do not work. The corresponding LRS and GLRS codes might hence be of interest for code-based cryptography.

Ever since their first mention, error correcting codes have played an important role in modern communication. Over the last few decades, they have gained in importance due to the ever-increasing amount of data that we store and communicate in our daily lives. To construct efficient error correcting codes, and their corresponding decoding algorithms, various mathematical tools from finite algebra, geometry, group theory, combinatorics, and many more, have proven extremely useful. This special issue provides a non-exhaustive overview of recent advances in algebraic coding theory, with a particular focus on the algebraic and geometric aspects of codes over finite fields. We want to emphasize this interdisciplinary connection, since many good error correcting codes have interesting algebraic or geometrical counterparts, which gives rise to powerful tools for classifying, characterizing, and investigating certain types of codes. The fourteen contributions to this special issue cover a broad range of topics, in particular subspace codes, quantum codes, rank metric codes, convolutional codes, self-dual and linear complementary dual codes, as well as codes related to finite projective geometries or finite groups, and finally, code-based cryptography. We express our deep gratitude to the authors of all the papers for their valuable contributions and to the reviewers for their accurate and timely work. We would like to thank Sihem Mesnager and Jintai Ding, Editors-in-Chief of the journal, for their support at all stages of the submission process.

We introduce a new weight and corresponding metric over finite extension fields for asymmetric error correction. The weight distinguishes between elements from the base field and the ones outside of it, which is motivated by asymmetric quantum codes. We set up the theoretic framework for this weight and metric, including upper and lower bounds, asymptotic behavior of random codes, and we show the existence of an optimal family of codes achieving the Singleton-type upper bound.