Let $C_g$ be a general curve of genus $g$. If $g\ge 4$ then the monodromy group of a primitive cover $C_g\to {\mathrm{\mathbb{P}}}^1$ of degree $n$ is either $S_n$ or $A_n$, and both cases actually occur (under suitable conditions on $n$ for fixed $g$). For $g=3$ also the groups $G{L}_3\left(2\right)$ and $AG{L}_3\left(2\right)$ occur. In the present paper we settle the last possible case of $AG{L}_4\left(2\right)$. This requires new methods (which may be of independent interest) studying the combinatorial structure of degenerate covers.

This note gives an explicit proof that the scalar subgroup of the Clifford-Weil group remains unchanged when passing to the quotient representation filling a gap in [3]. For other current and future errata to [3] see http://www.research.att.com/∼njas/doc/cliff2.html/.

Recently, CT burst errors originally defined for block codes have been generalized to CT burst errors for array codes [6]. In order to establish a Rieger’s type bound for array codes with respect to CT burst errors. Here, we introduce a CT burst error weight enumerator whose coefficients represent the number of CT burst errors of a particular weight. The method of obtaining the CT burst error weight enumerator is obtained by generating function like approach and it does not involve solving equations as presented in [6].

Finite group rings carry a natural involution that defines a form ring structure. We investigate the associated Clifford-Weil groups for the indecomposable representations of the groups of order $2,3$ and the symmetric group ${\mathrm{Sym}}_3$ over the fields with $2$ and $3$ elements as well as suitable symmetrizations. An analogue of Kneser’s neighboring method is introduced, to classify all self-dual codes in a given representation.

In previous work we determined automorphism groups of cyclic algebraic curves defined over fields of any odd characteristic. In this paper we determine parametric equations of families of curves for each automorphism group for such curves.

In this paper we describe a new method of encryption that originates from the public key cryptography and number theory. Our algorithm was inspired by the RSA algorithm and Diffie-Hellman key exchange protocol. It is based on a computationally difficult problem - the discrete logarithm problem in multiplicative group.

A method is given for the construction of linear codes with prescribed minimum distance and also prescribed minimum distance of the dual code. This works for codes over arbitrary finite fields. In the case of binary codes Matsumoto et al. showed how such codes can be used to construct cryptographic Boolean functions. This new method allows to compute new bounds on the size of such codes, extending the table of Matsumoto et al..

In this paper we will evaluate degrees of nonlinear polynomial encryption transformation in $F_q^n$, which was defined in [11] in terms of the walk on the graph with vertex set $F_q^n\cup F_g^n$. Independently from the length of the walk, this transformation has degree $3$. It means that public user can do the encryption process in polynomial time.

In this survey we study the genus $2$ curves with $\left(n,n\right)$-split Jacobian for even $n$.

Families of simple graphs of high girth had been used for the development of algorithms in Cryptography and Turbocoding. Recent results in that directions show the interest of applied researchers to ”families of directed graphs of high girth”, but the concept of the girth for the directed graphs is not well established. We discuss one of the possible definition. It agrees well with the classical definition in the case of simple graph and allows to create the analog of Extremal graph theory for simple graphs without small cycles for the class of balanced graphs i.e. directed graphs without multiple arrows such that each vertex has same number of inputs and outputs. Finally we discussed some explicit construction of simple and directed graphs which can be applicable to Turbocoding and Cryptography.

Any communication assumes a preliminary agreement between the parties involved. In our paper we address the question: what can we get when there is no agreement between the parties in the framework of classical communication quantum channels. We admit the concept of mixed coding and starting from it derive an idealized communication scheme based on continuous coding.