Double Circulant Matrices

Double circulant matrices are introduced and studied. A formula to compute the rank r of a double circulant matrix is exhibited; and it is shown that any consecutive r rows of the double circulant matrix are linearly independent. As a generalization, multiple circulant matrices are also introduced. Two questions on square double circulant matrices are suggested

Galois Self-Dual Constacyclic Codes

Generalizing Euclidean inner product and Hermitian inner product, we introduce Galois inner products, and study the Galois self-dual constacyclic codes in a very general setting by a uniform method. The conditions for existence of Galois self-dual and isometrically Galois self-dual constacyclic codes are obtained. As consequences, the results on self-dual, iso-dual and Hermitian self-dual constacyclic codes are derived.Comment: Key words: Constacyclic code, Galois inner product, $q$-coset function, isometry, Galois self-dual cod

Quasi-cyclic Codes of Index 1.5

We introduce quasi-cyclic codes of index 1.5, construct such codes in terms of polynomials and matrices; and prove that the quasi-cyclic codes of index 1.5 are asymptotically good

Permutation-like Matrix Groups with a Maximal Cycle of Length Power of Two

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [4], [5] and [6] showed that, if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals to a prime, or a square of a prime, or a power of an odd prime, then the permutation-like matrix group is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals to any power of 2, then it is similar to a permutation matrix group

Permutation-like Matrix Groups with a Maximal Cycle of Power of Odd Prime Length

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [4] and [5] showed that, if a permutation-like matrix group contains a maximal cycle of length equal to a prime or a square of a prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle of length equal to any power of any odd prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group

Self-dual Permutation Codes of Finite Groups in Semisimple Case

The existence and construction of self-dual codes in a permutation module of a finite group for the semisimple case are described from two aspects, one is from the point of view of the composition factors which are self-dual modules, the other one is from the point of view of the Galois group of the coefficient field.Comment: The main results of the manuscript have been published in DCC listed below, but the manuscript contains some more detailed annalysis and argument

Hyperbolic Modules of Finite Group Algebras over Finite Fields of Characteristic Two

Let $G$ be a finite group and let $F$ be a finite field of characteristic $2$. We introduce \emph{$F$-special subgroups} and \emph{$F$-special elements} of $G$. In the case where $F$ contains a $p$th primitive root of unity for each odd prime $p$ dividing the order of $G$ (e.g. it is the case once $F$ is a splitting field for all subgroups of $G$), the $F$-special elements of $G$ coincide with real elements of odd order. We prove that a symmetric $FG$-module $V$ is hyperbolic if and only if the restriction $V_D$ of $V$ to every $F$-special subgroup $D$ of $G$ is hyperbolic, and also, if and only if the characteristic polynomial on $V$ defined by every $F$-special element of $G$ is a square of a polynomial over $F$. Some immediate applications to characters, self-dual codes and Witt groups are given

Nonlinear functions and difference sets on group actions

Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonlinear functions and $G$-bent functions. Several known results in [2,6,10,17] are direct consequences of our results

Iso-Orthogonality and Type II Duadic Constacyclic Codes

Generalizing even-like duadic cyclic codes and Type-II duadic negacyclic codes, we introduce even-like (i.e.,Type-II) and odd-like duadic constacyclic codes, and study their properties and existence. We show that even-like duadic constacyclic codes are isometrically orthogonal, and the duals of even-like duadic constacyclic codes are odd-like duadic constacyclic codes. We exhibit necessary and sufficient conditions for the existence of even-like duadic constacyclic codes. A class of even-like duadic constacyclic codes which are alternant MDS-codes is constructed

Permutation-like Matrix Groups with a Maximal Cycle of Prime Square Length

A matrix group is said to be permutation-like if any matrix of the group is similar to a permutation matrix. G. Cigler proved that, if a permutation-like matrix group contains a normal cyclic subgroup which is generated by a maximal cycle and the matrix dimension is a prime, then the group is similar to a permutation matrix group. This paper extends the result to the case where the matrix dimension is a square of a prime