A class of generalized cyclic codes

1. IntroductionPrange was considered to be first person to study cyclic codes at the end of 1950s, seeIll and [2]. Since then, cyclic codes are the mostly studied of all codes, because they are easyto encode, and include an important family of BCH codes. A code C is cyclic if it is linearand if any cyclic shift of a codeword is also a codeword, i.e., whenever (co, of,' 1 on--l ) is inC then so is (c.--1, co j', c.--2). In fact, one could define a cyclic code to be an ideal in thering of…     

Some new quantum MDS codes from generalized Reed–Solomon codes

Quantum maximum distance separable (MDS) codes form a significant class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed–Solomon codes, we construct two new classes of $q$-ary quantum MDS codes, which have minimum distance greater than $\frac{q}{2}$. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.     

Construction of deletion correcting codes using generalized Reed–Solomon codes and their subcodes

A code is n-deletion correcting if it is possible to correct any n deletion of symbols having occurred in transmission of codewords. In this paper, we present explicit constructions of n-deletion correcting codes for arbitrary values of n using generalized Reed–Solomon codes and their subcodes.      

A class of binary cyclic codes with generalized Niho exponents

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined.     

Note on generalized Archimedes–Borchardt algorithm

Aequationes mathematicae - We investigate convergence and invariance properties of the generalized Archimedes–Borchardt algorithm. The main tool is reducing the problem to an appropriate...     

A note on the Assmus–Mattson theorem for some binary codes

Designs, Codes and Cryptography - We previously proposed the first nontrivial examples of a code having support t-designs for all weights obtained from the Assmus–Mattson theorem and having...     

Families of twisted tensor product codes

Using geometric properties of the variety ${\mathcal V_{r,t}}$ , the image under the Grassmannian map of a Desarguesian (t ? 1)-spread of PG(rt ? 1, q), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We determine the precise parameters of these codes and characterise the words of minimum weight.     

Markoff–Rosenberger triples and generalized Lucas sequences

Periodica Mathematica Hungarica - We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ with $$(x,y,z)=(U_i,U_j,U_k)$$ , where $$U_i$$ denotes the...     

A note on generalized twisted anomaly cancellation formulas

In this note, by studying modular invariance properties of some characteristic forms, we get some new twisted anomaly cancellation formulas.     

A class of finitary codes

It is shown that for any two Bernoulli schemes with a finite number of states and unequal entropies, there exists a finitary homomorphism from the scheme with larger entropy to the one with smaller entropy.     

On the persistence and blow up for the generalized two-component Dullin–Gottwald–Holm system

Considered herein is the persistence property of the solutions to the generalized two-component integrable Dullin–Gottwald–Holm system, which was derived from the Euler equation with nonzero constant vorticity in shallow water waves moving over a linear shear flow. Firstly, the persistence properties of the system are investigated in weighted $$L^p$$-spaces for a large class of moderate weights. Then, we establish the new local-in-space blow-up results simplifying and extending earlier blow-up criterion for this system.     

A class of almost MDS codes

MDS codes and almost MDS (AMDS) codes are special classes of linear codes, and have important applications in communications, data storage, combinatorial theory, and secrete sharing. The objective of this paper is to present a class of AMDS codes from some BCH codes and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes. The parameters of the duals of this class of AMDS codes are also discussed.     

A generalized extension theorem for linear codes

We prove that an [n, k, d] _q code \({\mathcal{C}}\) with gcd(d, q) = 1 is extendable if \({\sum_{i \not\equiv 0,d}A_i < (q-1)q^{k-2}}\), where A _i denotes the number of codewords of \({\mathcal{C}}\) with weight i. This is a generalization of extension theorems for linear codes by Hill and Lizak (Proceedings of the IEEE International Symposium on Information Theory, Whistler, Canada, 1995) and by Landjev and Rousseva (Probl. Inform. Transm. 42: 319–329, 2006).     

A Nash–Moser approach for the Euler–Arnold equations

We study the local well-posedness in the smooth category for a class of Euler equations. A Nash–Moser approach is used to extend, for the case of an invertible elliptic pseudo-differential operator, some results obtained by Escher and Kolev, with the help of some geometric arguments.     

A class of authentication codes with secrecy

We study a class of authentication codes with secrecy. We determine the maximum success probabilities of the impersonation and the substitution attacks on these codes and the level of secrecy. Therefore we give an answer to an open problem stated in Ding et al. (J Pure Appl Algebra 196:149–168, 2005). Our proofs use the number of rational places of a certain class of algebraic function fields. We determine this number by extending the corresponding results of E. Çakçak and F. Özbudak (Finite Fields Appl 14(1):209–220, 2008). Our authentication codes use a map which is not perfect nonlinear in certain subcases. We give an extended and unified approach so that the parameters of our authentication codes are good also when the corresponding map is not perfect nonlinear.