We study the weight distributions of cosets of the first-order Reed–Muller code (1 ) for odd , whose minimum weight is greater than or equal to the so-called quadratic bound. Some general restrictions on the weight distribution of a coset of (1 ) are obtained by partitioning its words according to their weight divisibility. Most notably, we show that there are exactly five weight distributions ...

Suppose q is a prime power and f ∈ Fq[x] is a univariate polynomial with exactly t nonzero terms and degree <q−1. To establish a finite field analogue of Descartes’ Rule, Bi, Cheng and Rojas (2013) proved an upper bound of 2(q − 1) t−2 t−1 on the number of cosets in F∗ q needed to cover the roots of f in F∗ q . Here, we give explicit f with root structure approaching this bound: For q a t power...

The q-Legendre polynomials can be treated as some special ”functions in the quantum double cosets U(1) \ SUq(2)/U(1)”. They form a family (depending on a parameter q) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense ”q-commutative”, the latter one is related t...

Let p be a prime number and F an algebraic closure of the finite field Fp with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations of the Galois group GQ = Gal(Q̄/Q) into GL(n,F), unramified at all finite primes not dividing pN . (We shall say “ρ is unramified outside pN .”) In this paper, “representation” will always mean “continuous, semisimple ...

In this study, nearness near-ring, subnearness M-group and ideal are introduced. By considering operations on the set of all near left weak cosets, near-ring cosets homomorphism also presented. Moreover, some properties these structures investigated.