On the differential uniformity of polynomials over Galois rings

On the groups of units of finite commutative chain rings

There have been several recent constructions of partial difference sets (PDSs) using the Galois rings GR(p^2, t) for p a prime and t any positive integer. This paper presents constructions of partial difference sets in (Z_{p^r})^{2t} where p is any prime, and r and t are any positive integers. For the case where r > 2 many of the partial difference sets are constructed in groups with parameters distinct from other known constructions, and the PDSs are nested. Another construction of Paley partial difference sets is given for the case when p is odd. The constructions make use of character theory and of the structure of the Galois ring GR(p^r, t), and in particular, the ring GR(p^r, t) \times GR(p^r, t). The paper concludes with some open related problems.

Cyclic [formula omitted]-additive codes

An explicit expression for Euclidean self-dual cyclic codes of length 2k over Galois ring GR(4,m)

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including $${\mathbb{Z}_{p^e}}$$ ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 e , l) of length n = 2 l for any a ? 1 and l ? 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.

Negacyclic codes of length 2 s over the Galois ring GR(2 a , m) are ideals of the chain ring \( \frac{{GR \left( {2^a ,m} \right)\left[ x \right]}} {{\left\langle {x^{2^s } + 1} \right\rangle }} \). This structure is used to provide the Hamming and homogeneous distances of all such negacyclic codes. The technique is then generalized to obtain the structure and Hamming and homogeneous distances of all γ-constacyclic codes of length 2 s over GR(2 a , m), where γ is any unit of the ring GR(2 a , m) that has the form γ = (4k 0 −1)+4k 1ξ +...+ 4k m−1ξ m−1, for integers k 0, k 1, ..., k m−1. Among other results, duals of such γ-constacyclic codes are studied, and necessary and sufficient conditions for the existence of a self-dual γ-constacyclic code are established.

Value sets of Dickson polynomials over Galois rings

Galois rings and exponential sums over Galois rings have many applications in algebraic combinatorics, coding theory and cryptography. In this paper, we present explicit description on the Gauss sums and Jacobi sums over Galois ring GR(p 2, r), and show that the values of these sums can be reduced to the Gauss sums and Jacobi sums over finite field \(\mathbb{F}_{p^r }\) for all non-trivial cases.

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. We study lattice isomorphisms of finite commutative rings with identity. The objective is to specify sufficient conditions subject to which rings under lattice homomorphisms preserve the following properties: to be a commutative ring, to be a ring with identity, to be decomposable into a direct sum of ideals. We look into the question about the projective image of the Jacobson radical of a ring. In the first part, the previously obtained results on projections of finite commutative semiprime rings are supplemented with new information. Lattice isomorphisms of finite commutative rings decomposable into direct sums of fields and nilpotent ideals are taken up in the second part. Rings definable by their subring lattices are exemplified. Projections of finite commutative rings decomposable into direct sums of Galois rings and nilpotent ideals are considered in the third part. It is proved that the presence in a ring of a direct summand definable by its subring lattice (i.e., the Galois ring GR(pn,m), where n > 1 and m > 1) leads to strong connections between the properties of R and R′.

Polynomials and primitive elements in Galois rings

Several new series of approximately mutually unbiased bases are constructed by using Gauss sums and Jacobi sums over Galois rings GR(p 2, r), and the tensor method.

Berger and Charpin (see ibid., vol.42, p.2194-2209, 1996 and Des., Codes Cuyptogr., vol.18, no.1/3, p.29-53, 1999) devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes. We extend this method to cyclic and extended cyclic codes over the Galois ring GR (p/sup a/, m), developing a generalization of the Mattson-Solomon polynomial. In particular, we classify all affine-invariant codes of length 2/sup m/ over Z/sub 4/, thus generalizing the corresponding result of Kasami, Lin, and Peterson (1967) and giving an alternative proof to Abdukhalikov. We give a large class of codes over Z/sub 4/ with large permutation groups, which include generalizations of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Muller (RM) codes.

Results on the quasi-cyclicity of the Gray map image of a class of codes defined over the Galois ring GR(p2,m) are given. These results generalize some appearing in [8] for codes over the ring of integers modulo p2(pa prime). The ring of (truncated) Witt vectors is a useful tool in proving the main results.

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelian p-groups. As an example, we also use Galois ring GR(4,2) to construct a (96,20,4) difference set in Z_4\times Z_4\times Z_6.

Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length n of the code is odd have been characterized over the alphabet Zopf4, and furthermore, have been generalized to the case of the alphabet being a finite commutative chain ring. In this paper, we investigate negacyclic codes of length 2s over Galois rings. The structure of negacyclic codes of length 2s over the Galois rings GR(2a,m), as well as that of their duals, are completely obtained. The Hamming distances of negacyclic codes over GR(2a,m) in general, and over Zopf2 a in particular are studied. Among other more general results, the Hamming distances of all negacyclic codes over Zopf2 a of length 4,8, and 16 are given. The weight distributions of such negacyclic codes are also discussed