8,221 research outputs found
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, -coset
function, isometry, Galois self-dual cod
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 be a finite group and let be a finite field of characteristic
. We introduce \emph{-special subgroups} and \emph{-special elements}
of . In the case where contains a th primitive root of unity for each
odd prime dividing the order of (e.g. it is the case once is a
splitting field for all subgroups of ), the -special elements of
coincide with real elements of odd order. We prove that a symmetric -module
is hyperbolic if and only if the restriction of to every
-special subgroup of is hyperbolic, and also, if and only if the
characteristic polynomial on defined by every -special element of is
a square of a polynomial over . Some immediate applications to characters,
self-dual codes and Witt groups are given
Nonlinear functions and difference sets on group actions
Let , be finite groups and let be a finite -set. -perfect
nonlinear functions from to have been studied in several papers. They
have more interesting properties than perfect nonlinear functions from
itself to . By introducing the concept of a -related difference
family of , we obtain a characterization of -perfect nonlinear functions
on . When is abelian, we characterize a -difference set of by the
Fourier transform on a normalized -dual set . We will also
investigate the existence and constructions of -perfect nonlinear functions
and -bent functions. Several known results in [2,6,10,17] are direct
consequences of our results
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
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
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