Given a field K of characteristic 2 and an integer n ≥ 2, let W (2n − 1, K) be the symplectic polar space defined in PG(2n − 1, K) by a nondegenerate alternating form of V (2n, K) and let Q(2n, K) be the quadric of PG(2n, K) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W (2n − 1, K) ∼= Q(2n, K). This is true when K is perfect, but false...

Let L(c, x) = e c √ log x log log x. We prove that if a 1 (mod q 1), ..., a k (mod q k) are a maximal collection of non-intersecting arithmetic progressions, with 2 ≤ q 1 < q 2 < · · · < q k ≤ x, then x L(√ 2 + o(1), x) < k < x L(1/6 − o(1), x). In the case for when the q i 's are square-free, we obtain the improved upper bound k < x L(1/2 − o(1), x) .

For fixed positive integers k, q, r with q a prime power and large m, we investigate matrices with m rows and a maximum number Nq(m, k, r) of columns, such that each column contains at most r nonzero entries from the finite field GF (q) and each k columns are linearly independent over GF (q). For even integers k ≥ 2 we obtain the lower bounds Nq(m, k, r) = Ω(m kr/(2(k−1))), and Nq(m, k, r) = Ω(...

We study geometric parameters associated with the Banach spaces (IRn, ‖·‖k,q) normed by ‖x‖k,q = (∑ 1≤i≤k |〈x, ai〉| )1/q , where {ai}i≤N is a given sequence of N points in IRn, 1 ≤ k ≤ N , 1 ≤ q ≤ ∞ and {λi }i≥1 denotes the decreasing rearrangement of a sequence {λi}i≥1 ⊂ IR. ∗

Geometric puncturing is a method to construct new codes from a given [n, k, d]q code by deleting the coordinates corresponding to some geometric object in PG(k − 1, q). We construct [gq(4, d), 4, d]q and [gq(4, d)+1, 4, d]q codes for some d by geometric puncturing, where gq(k, d) = ∑k−1 i=0 ⌈

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = n ∑ k=0 akx k ∈ K[x], an 6= 0. For p ∈ K[x]\F [x], define DF (p), the F deficit of p, to equal n −max{0 ≤ k ≤ n : ak / ∈ F}. For p ∈ F [x], define DF (p) = n. Let p(x) = n ∑ k=0 akx k, q(x) = m ∑ j=0 bjx j , with an 6= 0, bm 6= 0, an, bm ∈ F , bj / ∈ F for some j ≥ 1. Suppose ...

Introduction. Fix a base field k. The quantized coordinate ring of n×n matrices over k, denoted by q(Mn(k)), is a deformation of the classical coordinate ring of n×n matrices, (Mn(k)). As such, it is a k-algebra generated by n2 indeterminates Xij , for 1 ≤ i,j ≤ n, subject to relations which we state in (1.1). Here, q is a nonzero element of the field k. When q = 1, we recover (Mn(k)), which is...