A subset of n tuples of elements of Z9 is said to be a code over Z9 if it is a Z9-module. In this paper we consider an special family of cyclic codes over Z9, namely quadratic residue codes. We define these codes in term of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over finit...

We use conference matrices to define an action of the complex numbers on the real Euclidean vector space R. In certain cases, the lattice D n becomes a module over a ring of quadratic integers. We can then obtain new unimodular lattices, essentially by multiplying the lattice D n by a non-principal ideal in this ring. We show that lattices constructed via quadratic residue codes, including the ...

By combining Modified quadratic congruence (MQC) codes with M-sequence codes, a hybrid MQC/M-matrices coding over spectral and spatial domains is proposed for spectral-amplitude coding optical code-division multiple-access network. The system performance evaluation shows better result than the one of the previous system using M-matrices coding scheme.

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer n ≪ p1/2+ε, such that n! is a primitive root modulo p. We also show that every nonzero congruence ...

Introduced by Kraitchik and Lehmer, an x-pseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least x-pseudosquare that improves on a bound that is exponential in x due to Schinzel. We also obtain an equi-distribution result for pseudosquares. An x-pseudopower to base g is a posit...

Let F q be a finite field of order q with q = p n , where p is a prime. A multiplicative character χ is a homomorphism from the multiplicative group F * q , ·· to the unit circle. In this note we will mostly give a survey of work on bounds for the character sum x χ(x) over a subset of F q. In Section 5 we give a nontrivial estimate of character sums over subspaces of finite fields. §1. Burgess'...

For infinitely many primes p = 4k+1 we give a slightly improved upper bound for the maximal cardinality of a set B ⊂ Zp such that the difference set B−B contains only quadratic residues. Namely, instead of the ”trivial” bound |B| ≤ √p we prove |B| ≤ √ p − 1, under suitable conditions on p. The new bound is valid for approximately three quarters of the primes p = 4k + 1.

We will assume that the twisted L-series are primitive (i.e. the corresponding twisted modular forms are newforms). There is no loss of generality in making this assumption: if this is not the case, then f would be a quadratic twist of a newform of smaller level, which we can choose instead. The question of efficiently computing the family of central values L(f,D, 1), for fundamental discrimina...