Schur rings over a Galois ring of odd characteristic

It is proved that any Schur ring over a Galois ring of odd characteristic is either normal, or of rank 2, or a non-trivial generalized wreath product. The normal Schur rings are characterized as a special subclass of the cyclotomic Schur rings.     

Orthogonal graphs of odd characteristic and their automorphisms

The (isotropic) orthogonal graph O(2ν+δ,q) over of odd characteristic, where ν1 and δ=0,1 or 2 is introduced. When ν=1, O(21+δ,q) is a complete graph. When ν2, O(2ν+δ,q) is strongly regular and its parameters are computed, as well as its chromatic number. The automorphism groups of orthogonal graphs are also determined.     

Difference sets over Galois rings with odd extension degrees and characteristic an even power of 2

We construct an infinite family of (2 ^ns , 2 ^{ns/2 -1}(2 ^ns/2?1), 2 ^{ns/2 -1}(2 ^{ns/2 -1} ?1)) difference sets over a Galois ring GR(2 ⁿ , s) with characteristic an even power n of 2 and an odd extension degree s. It makes a chain of difference sets preserving the structures when n increases and s is fixed. We introduce a new operation into GR(2 ⁿ , s). The Gauss sum associated with the multiplicative character defined by the subgroup with respect to the new operation plays an important role in the construction.     

Negacyclic codes over Galois rings of characteristic 2 a

We investigate negacyclic codes over the Galois ring GR(2 a ,m) of length N = 2 k n,where n is odd and k 0.We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR(2 a ,m)[u]/ u 2 k + 1 .Then using a ring isomorphism we obtain the structure of negacyclic codes over GR(2 a ,m) of length N = 2 k n (n odd) and explore the existence of self-dual negacyclic codes over GR(2 a ,m).A bound for the homogeneous distance of such negacyclic codes is also given.     

Regular odd rings and non-planar graphs

In a previous paper we have announced that a graph is non-planar if and only if it contains a maximal, strict, compact, odd ring. Little has conjectured that the compactness condition may be removed. Chernyak has now published a proof of this conjecture. However, it is difficult to test a ring for maximality. In this paper we show that for odd rings of size five or greater, the condition of maximality may be replaced by a new one called regularity. Regularity is an easier condition to diagnose than is maximality.     

Subconstituents of orthogonal graphs of odd characteristic

In this paper, the subconstituents of the isotropic orthogonal graphs over finite fields of odd characteristic are studied. The first subconstituent is strongly regular, while the second subconstituent is edge-regular. The full automorphism groups of these two subconstituents have also been determined.     

Skew quasi-cyclic codes over Galois rings

Recently there has been a lot of interest on algebraic codes in the setting of skew polynomial rings. In this paper we have studied skew quasi-cyclic (QC) codes over Galois rings. We have given a necessary and sufficient condition for skew cyclic codes over Galois rings to be free, and determined a distance bound for free skew cyclic codes. A sufficient condition for 1-generator skew QC codes to be free is determined. Some distance bounds for free 1-generator skew QC codes are discussed. A canonical decomposition of skew QC codes is presented.     

Linear recurring sequences over Galois rings

Linear recurrences of maximal period over a Galois ring and over a residue class ring modulo p are studied. For any such recurrence, the coordinate sequences (in p-adic and some other expansions) are considered as linear recurring sequences over a finite field. Upper and lower bounds for the ranks (linear complexities) of these coordinate sequences are obtained. The results are based on using the properties of Galois rings and the trace-function on such rings.Translated fromAlgebra i Logika, Vol. 34, No. 2, pp. 169–189, March-April, 1995.     

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let be a Galois ring of characteristic 4,X a cyclotomic scheme over with classD and the Terwilliger algebra ofX, then we show that most of the irreducible -modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible -modules using Jacobi sums.     

Cardinal rank metric codes over Galois rings

In 1985, Gabidulin introduced the rank metric in coding theory over finite fields, and used this kind of codes in a McEliece cryptosystem, six years later. In this paper, we consider rank metric codes over Galois rings. We propose a suitable metric for codes over such rings, and show its main properties. With this metric, we define Gabidulin codes over Galois rings, propose an efficient decoding algorithm for them, and hint their cryptographic application.     

Galois theory over rings of arithmetic power series

Let R be a domain, complete with respect to a norm which defines a non-discrete topology on R. We prove that the quotient field of R is ample, generalizing a theorem of Pop. We then consider the case where R is a ring of arithmetic power series which are holomorphic on the closed disc of radius 0<r<1 around the origin, and apply the above result to prove that the absolute Galois group of the quotient field of R is semi-free. This strengthens a theorem of Harbater, who solved the inverse Galois problem over these fields.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

Finite Euclidean graphs over rings

We consider graphs attached to , where , for an odd prime , using an analogue of the Euclidean distance. These graphs are shown to be mostly non-Ramanujan, in contrast to the case of Euclidean graphs over finite fields.

     

Endomorphisms and cores of quadratic forms graphs in odd characteristic

A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. Let ${\mathbb{F}}_q$ be the finite field with q elements where q is a power of an odd prime number. The quadratic forms graph, denoted by $\mathrm{Quad}(n,q)$ where $n\ge 2$, has all quadratic forms on ${\mathbb{F}}_q^n$ as vertices and two vertices f and g are adjacent whenever $\mathrm{rk}(f-g)=1$ or 2. We prove that every $\mathrm{Quad}(n,q)$ is a pseudo-core. Further, when n is even, $\mathrm{Quad}(n,q)$ is a core. When n is odd, $\mathrm{Quad}(n,q)$ is not a core. On the other hand, we completely determine the independence number of $\mathrm{Quad}(n,q)$.