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Triorthogonal matrices were introduced in quantum information theory in connection with distillation of magic states (Bravyi and Haah in Phys Rev A 86:052329, 2012). We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. Further, we generalize to this setting the classical coding techniques of shortening and extending. We also give some simple propagation rules. [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]

We study codes with parameters of $q$ -ary shortened Hamming codes, i.e., $(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$ -fold packing with minimum distance 2. Secondly, for every admissible length starting from $n=20$ , we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code. [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]

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the $\mathbb {Z}_{2} \mathbb {Z}_{4}$ -additive quasi-cyclic codes. [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]

Mixed alphabet codes are the generalizations of the classical linear codes over finite fields and rings. In this letter we introduce a valid method of constructing FpR-additive codes, where p is a prime number, and R = Fp + uFp with u2 = 0. These codes can be viewed as a combination of irreducible cyclic codes over Fp and trace codes over R. In particular, by using the Gray map, part of the obtained image codes meets the Greismer bound with equality, and can be applied to construct new secret sharing scheme. [ABSTRACT FROM AUTHOR]

Two infinite families of Z 4 -codes with two nonzero Lee weights are constructed by their generator matrices. Their Gray images are nonlinear with the same weight distribution as that of the two-weight binary codes of type SU1 in the sense of (Calderbank, Kantor, 1986). [ABSTRACT FROM AUTHOR]

The dual of the Kasami code of length $q^{2}-1$ , with $q$ a power of 2, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_{q}$ with a Simplex code of length $q-1$. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related $q$ -ary MDS code are determined. New cyclic completely regular codes over finite fields a power of 2, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code. [ABSTRACT FROM AUTHOR]

The dodecacode is a nonlinear additive quaternary code of length 12. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance 5. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on 2 10 vertices, with new intersection array { 33 , 30 , 15 ; 1 , 2 , 15 } . The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters (2 10 , 495 , 238 , 240) . Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length 33 is constructed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Toeplitz codes of index t , which are a generalization of quasi-cyclic codes of index t. We show that such codes meet the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 1 t over F q , so they are asymptotically good for any t. We characterize some special classes of Toeplitz codes as LCD codes (resp. linear codes with one-dimensional hull or self-orthogonal codes). It is worth mentioning that we determine the hull dimension of the isometric image of a linear code. Some optimal or almost optimal binary linear codes are obtained as a result. Most importantly, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) from quasi-cyclic codes. In particular, we construct binary LCD [ 32 , 16 , 8 ] , [ 33 , 11 , 11 ] and [ 34 , 17 , 8 ] codes with improved parameters directly. [ABSTRACT FROM AUTHOR]

In this paper, two different Gray-like maps from $Z_{p}^\alpha \times Z_{p^{k}}^\beta $ , where $p$ is prime, to $Z_{p}^{n}$ , $n={\alpha +\beta p^{k-1}}$ , denoted by $\varphi $ and $\Phi $ , respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_{p} Z_{p^{k}}$ -additive code, and $C^\bot $ is its dual, then the weight enumerators of the image $p$ -ary codes $\varphi (C)$ and $\Phi (C^\bot)$ are formally dual. This is a partial generalization of [D. S. Krotov, On $Z_{2^{k}}$ -dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532–1537], and the result is generalized to odd characteristic $p$ and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed $Z_{p} Z_{p^{2}}\ldots Z_{p^{k}}$ alphabet is given. [ABSTRACT FROM AUTHOR]

We construct an infinite family of two-Lee-weight codes over the ring F 2 + u F 2 + v F 2 + u v F 2 . These codes are defined as trace codes and have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Then, taking Gray mapping, we obtain an infinite family of abelian binary two-weight codes which are shown to be optimal by application of the Griesmer bound. Moreover, an application to secret sharing schemes is given. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problems raised in [8] and present some examples to illustrate the obtained results. Moreover, we work out the diophantine problem by Shi and Wang and then give the sufficient conditions for the nonexistence of two-Lee weight projective codes over \mathbb Z4 with type 4^k12^{k2} . [ABSTRACT FROM PUBLISHER]

We construct an infinite family of two-Lee-weight codes over the ring \mathbb F2+u\mathbb F2 . These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. By Gray mapping, we obtain an infinite family of abelian binary two-weight codes. They are shown to be optimal by application of the Griesmer bound. An application to secret sharing schemes is given. [ABSTRACT FROM PUBLISHER]

An additive cyclic code of length n over F 4 can be defined equivalently as an F 2 [ x ] / 〈 x n + 1 〉 -submodule of F 4 [ x ] / 〈 x n + 1 〉. In this paper we study additive cyclic and complementary dual codes of odd length over F 4 with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of these codes by their generators as binary cyclic codes. [ABSTRACT FROM AUTHOR]

Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, (1 + u)-constacyclic codes over Z + uZ of any length are studied. A new Gray map between Z + uZ and Z is defined. By means of this map, it is shown that the Z Gray image of a (1 + u)-constacyclic code of length n over Z + uZ is a cyclic code over Z of length 4 n. Furthermore, by combining the classical Gray map between Z and F , it is shown that the binary image of a (1 + u)-constacyclic code of length n over Z + uZ is a distance invariant binary quasi-cyclic code of index 4 and length 8 n. Examples of good binary codes are constructed to illustrate the application of this class of codes. [ABSTRACT FROM AUTHOR]

Let R = Z 4 be the integer ring mod 4. A double cyclic code of length ( r , s ) over R is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . In this paper, we determine the generator polynomials of this family of codes as R [ x ] -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Further, we also give the minimal generating sets of this family of codes as R -submodules of R [ x ] / ( x r − 1 ) × R [ x ] / ( x s − 1 ) . Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual. [ABSTRACT FROM AUTHOR]

Double Toeplitz (DT) codes are codes with a generator matrix of the form (I , T) with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields. [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]