Negacyclic codes over Galois rings of characteristic 2a

Abstract We investigate negacyclic codes over the Galois ring GR（2a,m） of length N = 2kn, 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（2a, m）[u]/〈u2k ＋ 1〉. Then using a ring isomorphism we obtain the structure of negacyclic codes over GR（2a, m） of length N = 2kn （n odd） and explore the existence of self-dual negacyclic codes over GR（2a, m）. A bound for the homogeneous distance of such negacvclic codes is also given.

Constructions on Approximately Mutually Unbiased Bases by Galois Rings

Several new series of approximately mutually unbiased bases are constructed by using Gauss sums and Jacobi sums over Galois rings GR(p2, r), and the tensor method.

GMW sequences over Galois rings and their linear complexities

A new family of GMW sequences over an arbitrary Galois ring was defined by using the trace functions and permutations.This generalizes the concept of GMW sequences over finite fields.Utilizing the Fourier representation,we derived an estimate of the linear complexities of this family of GMW sequences.And the result shows that such sequences have large linear complexities.

The 0 ,1 - Distribution of the First CoordinateSequences Derived from ML- Sequences over Z/(2~e)

The 0,1 distribution of the first coordinate sequences derived from the maximal lenght sequences (ML sequences) over Z/(2~e) is exactly given.

A Construction of Relative Difference Set

In this paper, we give a construction of RDS in Galois ring by using some bent function, and obtain the equivalent relationship between RDS and a kind of bent function. At the same time, its existence is demonstrated.

Improved Information Set Decoding Algorithms over Galois Ring in the Lee Metric

The security of most code-based cryptosystems relies on the hardness of the syndrome decoding(SD) problem.The best solvers of the SD problem are known as information set,decoding(ISD) algorithms.Recently,Weger,et al.(2020) described Stern’s ISD algorithm,s-blocks algorithm and partial Gaussian elimination algorithms in the Lee metric over an integer residue ring Z_(pm),where p is a prime number and m is a positive integer,and analyzed the time complexity.In this paper,the authors apply a binary ISD algorithm in the Hamming metric proposed by May,et al.(2011)to solve the SD problem over the Galois ring GR(p^(m),k) endowed with the Lee metric and provide a detailed complexity analysis.Compared with Stern’s algorithm over Zpmin the Lee metric,the proposed algorithm has a significant improvement in the time complexity.

The Gauss sums and Jacobi sums over Galois ring GR(p^2 , r)

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(p2 , r), and show that the values of these sums can be reduced to the Gauss sums and Jacobi sums over finite field Fpr for all non-trivial cases.

The maximal size of 6-and 7-arcs in projective Hjelmslev planes over chain rings of order 9 Dedicated to Professor Feng Keqin on the Occasion of his 70th Birthday

We complete the determination of the maximum sizes of （k,n）-arcs, n ≤ 12, in the projective gjelmslev planes over the two （proper） chain rings Z9 = Z/9Z and S3 = F3[X]/（X2） of order 9 by resolving the hitherto open cases n = 6 and n = 7. Parts of our proofs rely on decidedly geometric properties of the planes such as Desargues＇ theorem and the existence of certain subplanes.

Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe,d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x-1)m+pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe,d) is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.

0,1 distribution in the highest level sequences of primitive sequences over Z2e

In this paper, we discuss the 0, 1 distribution in the highest level sequence ae-1 of primitive sequence over Z2e generated by a primitive polynomial of degree n. First we get an estimate of the 0, 1 distribution by using the estimates of exponential sums over Galois rings, which is tight for e relatively small to n. We also get an estimate which is suitable for e relatively large to n. Combining the two bounds, we obtain an estimate depending only on n, which shows that the larger n is, the closer to 1/2 the proportion of 1 will be.