Smooth lattices over quadratic integers

We construct lattices with quadratic structure over the integers in quadratic number fields having the property that the rank of the quadratic structure is constant and equal to the rank of the lattice in all reductions modulo maximal ideals. We characterize the case in which such lattices are free. The construction gives a representative of the genus of such lattices as an orthogonal sum of “standard” pieces of ranks 1–4 and covers the case of the discriminant of the real quadratic number field congruent to 1 modulo 8 for which a general construction was not known.      

Goppa codes over the p-adic integers and integers modulo pe

Goppa codes were defined by Valery D. Goppa in 1970. In 1978, Robert J. McEliece used this family of error-correcting codes in his cryptosystem, which has gained popularity in the last decade due to its resistance to attacks from quantum computers. In this paper, we present Goppa codes over the p-adic integers and integers modulo $p^e$. This allows the creation of chains of Goppa codes over different rings. We show some of their properties, such as parity-check matrices and minimum distance, and suggest their cryptographic application, following McEliece's scheme.     

Complexity of optimizing over the integers

Mathematical Programming - In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete...     

Codes over Zm

In this paper we study cyclic codes inZ _m. i.e., ideals inZ _mG, G a finite abelian group, and we give a classification of such codes.We also study the minimum Hamming distance and the generalized Hamming weight of BCH codes overZ _m.     

On Ramanujan sums over the Gaussian integers

This note deals with Ramanujan sums c _m (n) over the ring ?[i], in particular with asymptotics for sums of c _m (n) taken over both variables m, n.     

Separable free quadratic algebras over quadratic integers

The aim of the paper is to determine all free separable quadratic algebras over the rings of integers of quadratic fields in terms of the properties of the fundamental unit in the real case. The paper corrects some earlier published results on the subject.     

Z_(p~m)-线性码

给出了含非零码字的任一 Zpm-线性码的生成矩阵形式 (其中 p为素数 ,m为正整数 ) ,推广了文 [1 ]的结论     

Sums over positive integers with few prime factors

The author has presented estimates for sums of multiplicative functions, satisfying certain conditions, extended over positive integers n such that n is less than or equal to x^t, the greatest prime factor of n is at most equal to x, and n is relatively prime to a natural number k. These estimates are uniform in t, x, and k.     

Ideal membership in polynomial rings over the integers

We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials , where is an -tuple of indeterminates, are there such that ? We show that the degree of the polynomials can be bounded by where is the maximum total degree and the maximum height of the coefficients of . Some related questions, primarily concerning linear equations in , where is the ring of integers of a number field, are also treated.

     

Finitary groups and Krull dimension over the integers

Let M be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup

Weyl sums over integers with affine digit restrictions

For any given integer q?2, we consider sets N of non-negative integers that are defined by affine relations between their q-adic digits (for example, the set of non-negative integers such that the number of 1's equals twice the number of 0's in the binary representation). The main goal is to prove that the sequence (αn)_n∈N is uniformly distributed modulo 1 for all irrational numbers α. The proof is based on a saddle point analysis of certain generating functions that allows us to bound the corresponding Weyl sums.     

Prime ideals in two-dimensional domains over the integers

Let B be a finitely generated birational extension of , the ring of polynomials in one variable over the integers . (That is, B is a finitely generated extension of contained in its quotient field Then Spec(B) is order-isomorphic to Spec . This affirms part of a conjecture of Wiegand (1986).     

Projective semimodules over semirings with valuations in nonnegative integers

We introduce semirings with valuations in nonnegative integers and prove that all projective semimodules over them are free.