Projective two-weight irreducible cyclic and constacyclic codes

After several remarks on two-weight irreducible cyclic codes, we introduce a family of projective two-weight cyclic codes and a family of projective two-weight constacyclic codes and we discuss the existence of such codes.     

Quasi-cyclic codes as cyclic codes over a family of local rings

We give an algebraic structure for a large family of binary quasi-cyclic codes. We construct a family of commutative rings and a canonical Gray map such that cyclic codes over this family of rings produce quasi-cyclic codes of arbitrary index in the Hamming space via the Gray map. We use the Gray map to produce optimal linear codes that are quasi-cyclic.     

Some new non-additive maximum rank distance codes

In this paper, a construction of maximum rank distance (MRD) codes as a generalization of generalized Gabidulin codes is given. The family of the resulting codes is not covered properly by additive generalized twisted Gabidulin codes, and does not cover all twisted Gabidulin codes. When the basis field has more than two elements, this family includes also non-affine MRD codes, and such codes exist for all parameters. Therefore, these codes are the first non-additive MRD codes for most of the parameters.     

Symplectic spread-based generalized Kerdock codes

Kerdock codes (Kerdock, Inform Control 20:182–187, 1972) are a well-known family of non-linear binary codes with good parameters admitting a linear presentation in terms of codes over the ring (see Nechaev, Diskret Mat 1:123–139, 1989; Hammons et al., IEEE Trans Inform Theory 40:301–319, 1994). These codes have been generalized in different directions: in Calderbank et al. (Proc Lond Math Soc 75:436–480, 1997) a symplectic construction of non-linear binary codes with the same parameters of the Kerdock codes has been given. Such codes are not necessarily equivalent. On the other hand, in Kuzmin and Nechaev (Russ Math Surv 49(5), 1994) the authors give a family of non-linear codes over the finite field F of q = 2 ^l elements, all of them admitting a linear presentation over the Galois Ring R of cardinality q ² and characteristic 2². The aim of this article is to merge both approaches, obtaining in this way new families of non-linear codes over F that can be presented as linear codes over the Galois Ring R. The construction uses symplectic spreads.      

Some families of -cyclic codes

We introduce and solve several problems on -cyclic codes.We study the link between -linear cyclic codes and -cyclic codes (not necessarily linear) obtained by using two binary linear cyclic codes. We use these results to present a family of -self-dual linear cyclic codes.     

Diagonally neighbour transitive codes and frequency permutation arrays

Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication, while at the same time maintaining a constant power output. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family.     

A family of ternary quasi-perfect BCH codes

In this paper we present a family of ternary quasi-perfect BCH codes. These codes are of minimum distance 5 and covering radius 3. The first member of this family is the ternary quadratic-residue code of length 13.     

Polyadic codes of prime power length

Polyadic codes constitute a special class of cyclic codes and are generalizations of quadratic residue codes, duadic codes, triadic codes, m-adic residue codes and split group codes, which have good error-correcting properties. In this paper, we give necessary and sufficient conditions for the existence of polyadic codes of prime power length. Examples of some good codes arising from the family of polyadic codes of prime power length are also given.     

Classification of Extremal Double Circulant Formally Self-Dual Even Codes

Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes.     

A link between combinatorial designs and three-weight linear codes

In this paper, we show that partial geometric designs can be constructed from certain three-weight linear codes, almost bent functions and ternary weakly regular bent functions. In particular, we show that existence of a family of partial geometric difference sets is equivalent to existence of a certain family of three-weight linear codes. We also provide a link between ternary weakly regular bent functions, three-weight linear codes and partial geometric difference sets.     

Johnson type bounds on constant dimension codes

Very recently, an operator channel was defined by Koetter and Kschischang when they studied random network coding. They also introduced constant dimension codes and demonstrated that these codes can be employed to correct errors and/or erasures over the operator channel. Constant dimension codes are equivalent to the so-called linear authentication codes introduced by Wang, Xing and Safavi-Naini when constructing distributed authentication systems in 2003. In this paper, we study constant dimension codes. It is shown that Steiner structures are optimal constant dimension codes achieving the Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Then, we derive two Johnson type upper bounds, say I and II, on constant dimension codes. The Johnson type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known Steiner structures is actually a family of optimal constant dimension codes achieving both the Johnson type bounds I and II.      

Linear transformations on codes

This paper studies and classifies linear transformations that connect Hamming distances of codes. These include irreducible linear transformations and their concatenations. Their effect on the Hamming weights of codewords is investigated. Both linear and non-linear codes over fields are considered. We construct optimal linear codes and a family of pure binary quantum codes using these transformations.     

Codes over an infinite family of rings with a Gray map

Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.     

Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Designs, Codes and Cryptography - In this paper, we construct a family of non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Moreover, we present a...     

Tensor products of ideal codes over Hopf algebras

We study indecomposable codes over a family of Hopf algebras introduced by Radford. We use properties of Hopf algebras to show that tensors of ideal codes are ideal codes, extending the corresponding result that was previously given in the case of Taft Hopf algebras and showing the differences with that case.     

The weight distributions of some cyclic codes with three or four nonzeros over \mathbb F _3

Because of efficient encoding and decoding algorithms comparing with linear block codes, cyclic codes form an important family and have applications in communications and storage systems. However, their weight distributions are known only for a few cases mainly on the codes with no more than three nonzeros. In this paper, the weight distributions of two classes of cyclic codes with three or four nonzeros are determined.     

2n Bordered constructions of self-dual codes from group rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes of length 64 and 68, constructing 30 new extremal self-dual codes of length 68.     

ON THE CONSTRUCTION OF AUTHENTICATION CODES THAT PERMIT ARBITRATION OVER PROJECTIVE SPACES

A family of authentication codes with arbitration are constructed over projective spaces, the parameters and the probabilities of deceptions of the codes are also computed. In a special case, a perfect authentication code with arbitration is obtained.     

On two variations of identifying codes

Identifying codes have been introduced in 1998 to model fault detection in multiprocessor systems. In this paper, we introduce two variations of identifying codes: weak codes and light codes. They correspond to fault detection by successive rounds. We give exact bounds for those two definitions for the family of cycles.