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Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on m $m$‐by‐n $n$ matrices over a finite field Fq ${{\mathbb{F}}}_{q}$. They are diameter perfect and have the cardinality qm(n−d+1) ${q}^{m(n-d+1)}$ if m≥n $m\ge n$. We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in m $m$ if the other parameters (n,q $n,\,q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. [ABSTRACT FROM AUTHOR]

The hull of a linear code over finite fields is the intersection of the code and its dual code, which has been widely studied due to its wide applications. In this paper, we develop a general method for constructing linear codes with small hulls using the eigenvalues of the generator matrices. Using this method, we construct many optimal Euclidean and Hermitian LCD codes, which improve the previously known lower bound on the largest minimum distance. We also obtain many (near) MDS LCD codes and (near) MDS codes with one-dimensional hull. Furthermore, we give three tables about formally self-dual LCD codes. [ABSTRACT FROM AUTHOR]

A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have been a hot topic recently due to Boolean masking application in the security of embarked electronics (Carlet and Guilley in Pinto et al (eds) Coding theory and applications, Springer, CIMSMS, Berlin, 2015). Additive codes over F 4 are F 4 -codes that are stable by codeword addition but not necessarily by scalar multiplication. An additive code over F 4 is additive complementary dual (ACD) if it meets its dual trivially. The aim of this research is to study such codes which meet their dual trivially. All the techniques and problems used to study LCD codes are potentially relevant to ACD codes. Interesting constructions of ACD codes from binary codes are given with respect to the trace Hermitian and trace Euclidean inner product. The former product is relevant to quantum codes. [ABSTRACT FROM AUTHOR]

In this paper, we first generalize the polycyclic codes over finite fields to polycyclic codes over the mixed alphabet Z 2 Z 4 , and we show that these codes can be identified as Z 4 [ x ] -submodules of R α , β with R α , β = Z 2 [ x ] / ⟨ t 1 (x) ⟩ × Z 4 [ x ] / ⟨ t 2 (x) ⟩ , where t 1 (x) and t 2 (x) are monic polynomials over Z 2 and Z 4 , respectively. Then we provide the generator polynomials and minimal generating sets for this family of codes based on the strong Gröbner basis. In particular, under the proper defined inner product, we study the dual of Z 2 Z 4 -additive polycyclic codes. Finally, we focus on the characterization of the Z 2 Z 4 -MDSS and MDSR codes, and as examples, we also present some (almost) optimal binary codes derived from the Z 2 Z 4 -additive polycyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we give a new method to answer a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the equation x d + (x + 1) d = b in F q 4 , where n is a positive integer, q = 2 n and d = q 3 + q 2 + q − 1. In particular, we directly determine the differential spectrum of this power function x d using a more simple and direct method compared with those in the literature. [ABSTRACT FROM AUTHOR]

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7]. [ABSTRACT FROM AUTHOR]

We prove that the covering radius of the Melas code $M(m,q)$ of length $n=q^{m}-1$ over $\mathbb {F}_{q}$ is 2 if $q > 3$. We also prove that the covering radius of $M(m,3)$ is 3 is $m \ge 3$ , the covering radius of $M(2,3)$ is 4, and the covering radii of $M(1,2)$ and $M(1,3)$ are 1. [ABSTRACT FROM AUTHOR]

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code-division multiple-access (MC-CDMA) systems. They can support more users than perfect complementary sequence sets in MC-CDMA systems. It is desirable to design QCSSs with good parameters that are a trade-off of large set size, small periodic maximum magnitude correlation and small alphabet size. The main results are to construct new infinite families of QCSSs that all have small alphabet size and asymptotically optimal periodic maximum magnitude correlation. In this paper, we propose three new constructions of QCSSs using additive characters over finite fields. Notably, these QCSSs have new parameters and small alphabet sizes. Using the properties of characters and character sums, we determine their maximum periodic correlation magnitudes and prove that these QCSSs are asymptotically optimal with respect to the lower bound. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, code extension and projection over a self-dual basis are the main tools. Some optimal, almost MDS, and MDS self-dual codes over both small and large finite fields are constructed. Moreover, over fifty MDS codes with new parameters are constructed. Comparisons with classical constructions are made. [ABSTRACT FROM AUTHOR]

In this paper 1 , we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis. [ABSTRACT FROM AUTHOR]

In this work, we study a family of self-dual four circulant codes and a family of self-dual four negacirculant codes, and the exact counting formula is derived for these families of codes. In addition, we prove that asymptotically good self-dual four circulant codes and negacirculant codes over finite fields exist, and both of them satisfy a modified Gilbert-Varshamov bound on the relative minimum distance with asymptotic rate 1 2. [ABSTRACT FROM AUTHOR]