Codebooks with small maximal cross-correlation amplitudes have important applications in code division multiple access (CDMA) communication, coding theory and compressed sensing. In this letter, we design a new codebook based on construction of Ramanujan graphs over finite abelian groups. We prove that the length K=q+1 size N=q2+2q+2 is asymptotically optimal nearly achieving Levenshtein bound ...

Let S(n, 0) be the set of monic complex polynomials of degree n ≥ 2 having all their zeros in the closed unit disk and vanishing at 0. For p ∈ S(n, 0) denote by |p|0 the distance from the origin to the zero set of p. We determine all 0-maximal polynomials of degree n, that is, all polynomials p ∈ S(n, 0) such that |p|0 ≥ |q|0 for any q ∈ S(n, 0). Using a second order variational method we then ...

Input: An integer n > 1. 0: if n is a power then output composite fi; 1: r := 2; 2: while (r < n) do 3: if gcd(r, n) 6= 1 then output composite fi; 4: if r is prime then 5: q := largest prime factor of r − 1; 6: if (q ≥ 4√r log n) and (n(r−1)/q 6≡ 1 mod r) then break fi; 7: fi; 8: r := r + 1; 9: od; 10: for a = 1 to 2 √ r log n do 11: if (x− a)n 6≡ (xn − a) mod (xr − 1, n) then output composite...

One can apply Bonse’s inequality to points on the real line to find an upper bound on any prime gap, including both for first occurrences and for maximal prime gaps. However, such a result is neither as fine as the upper bound found by Mozzochi, nor as fine as the lower bound obtained by Rankin for maximal prime gaps. Without deep sieve methods, such as those used by Maier and Pomerance to comp...

Let f(n) or the base-2 logarithm of f(n) be either d(n) (the divisor function), σ(n) (the divisor-sum function), φ(n) (the Euler totient function), ω(n) (the number of distinct prime factors of n) or Ω(n) (the total number of prime factors of n). We present good lower bounds for ∣ ∣M N − α ∣ ∣ in terms of N , where α = [0; f(1), f(2), . . .].

A ring $A$ is (principally) nilary, denoted (pr-)nilary, if whenever $XY=0,$ then there exists a positive integer $n$ such that either $X^n=0$ or $Y^n=0$ for all (principal) ideals $X$, $Y$ of $A$. We determine necessary and/or sufficient conditions the group $A[G]$ to be nilary in terms on $G$. For example, we show that: (1) If (pr-)nilary and $G$ prime order each finite nontrivial normal subg...

Abstract Let k be a finite field of characteristic p , and G compact -adic analytic group. Write kG for the completed group ring over . In this paper, we describe structure / P where is minimal prime ideal We give an explicit isomorphism between matrix with coefficients in ${(k'G')_\alpha }$ $k'/k$ extension, $G'$ large subquotient no normal subgroups, (–) α “twisting” operation that preserves ...