Three bii linear codes of length 27 and 28 are described.They cuntaiu more vectors than any previously known codes with the samelength and miuimIm3 distance. Let gi(x) and g2(x) be the following polynomials over GF(2):

We consider the partition function bp(n), which counts the number of partitions of the integer n into distinct parts with no part divisible by the prime p. We prove the following: Let p be a prime greater than 3 and let r be an integer between 1 and p−1, inclusively, such that 24r + 1 is a quadratic nonresidue modulo p. Then, for all nonnegative integers n, bp(pn + r) ≡ 0 (mod 2).

We show that for any ε > 0 and a sufficiently large cube-free q, any reduced residue class modulo q can be represented as a product of 14 integers from the interval [1, q 1/2+ε]. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications t...

We show how to produce short proofs of theorems such that a distrusting Verifier can be convinced that the theorem is true yet obtains no information about the proof itself. The proofs are non-interactive provided that the quadratic residuosity bit commitment scheme is available to the Prover and Verifier. For typical applications, the proofs are short enough to fit on a floppy disk.

Traveling waves in one-dimensional nonlinear periodic structures are investigated for low-amplitude oscillations using perturbation analysis. We use second-order multiple scales analysis to capture the effect of quadratic nonlinearity. Comparisons with the linear and cubical nonlinear cases are presented in the dispersion relationship, group velocity and phase velocity and their dependence on w...

Factoring an integer is equivalent to express the integer as the difference of two squares. We test that for any odd modulus, in the corresponding ring of remainders, any element can be realized as the difference of two quadratic residues, and also that, for a fixed remainder value, the map assigning to each modulus the number of ways to express the remainder as difference of quadratic residues...

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest possible number of consecutive quadratic non-residues modulo p is always less than p. (One can find a brief discussion of this problem in R. K. Guy’s book [5]). Schur also pointed out that the greatest number of consecutive q...

Let A: be an integer > 2 andp a prime such that vk(p) = (k,p — 1) > 1. Let bn + c(n = 0,1,. ..;b > 2,1 < c < b, (b,p) — (c,p) = 1) be an arithmetic progression. We denote the smallest kth power nonresidue in the progression bn + c by g(p,k,b,c), the smallest quadratic residue in the progression bn + c by r2(p,b,c), and the nth smallest prime kth power nonresidue by g„(p,k), n = 0, 1, 2,_ If C(p...

We study several joint facility location and inventory management problems with stochastic retailer demand. In particular, we consider cases with uncapacitated facilities, capacitated facilities, correlated retailer demand, stochastic lead times, and multicommodities. We show how to formulate these problems as conic quadratic mixed-integer problems. Valid inequalities, including extended polyma...