The idea of q cyclotomic cosets modulo n due to carry Huffman and Vera pless is considered in the context of linear codes of length n over ( ) GF q where m q p  , p is a prime, 1 m  . An explicit formula for the number of q cyclostomes cosets modulo is obtained as an application of the classical Burnsides lemnra. The formula for the number of cyclic codes in   ( 1) n n q R F x x   is dedu...

We conjecture polynomial identities which imply Rogers–Ramanujan type identities for branching functions associated with the cosets (G)l−1⊗(G )1/(G )l, with G=An−1 (l ≥ 2), Dn−1 (l ≥ 2), E6,7,8 (l = 2). In support of our conjectures we establish the correct behaviour under level-rank duality for G=An−1 and show that the A-D-E Rogers–Ramanujan identities have the expected q → 1− asymptotics in t...

Let H and K be subgroups of a finite group G . Pick g ? uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) = Borel subgroup L n ( F q ) This leads to new theorems for Mallows measure permutations insights into LU matrix factorization. 2) The cosets hyperoctahedral inside S 2 , which applications Ewens's sampling formula mathematical...

We introduce a new approach for the study of weight distributions of cosets of the Reed–Muller code of order 1. Our approach is based on the method introduced by Kasami in [1], using Pless identities. By interpreting some equations, we obtain a necessary condition for a coset to have a “high” minimum weight. Most notably, we are able to distinguish such cosets which have three weights only. We ...

In this paper, we define the concept of a bipolar fuzzy cosets of a bipolar fuzzy HX subgroup, bipolar anti fuzzy HX subgroup, bipolar fuzzy middle coset of a bipolar fuzzy HX subgroup and bipolar anti fuzzy HX subgroup, conjugate bipolar fuzzy HX subgroup, conjugate bipolar anti fuzzy HX subgroup and discussed some of its properties with the examples. Further we define the level subsets and lo...

The history, definition and principal properties of automatic groups and their generalisations to subgroups and cosets are reviewed briefly, mainly from a computational perspective. A result about the asynchronous automaticity of an HNN extension is then proved and applied to an example that was proposed by Mark Sapir. AMS Classification 20F32; 20F05

New properties of q-ary cyclotomic cosets modulo n = qm − 1, where q ≥ 3 is a prime power, are investigated in this paper. Based on these properties, the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed. As an application, new families of nonbinary Calderbank-Shor-Steane (CSS) quantum codes as well as new families of convolutional ...

The number ofmoves required to solve any configuration of Rubik’s cube has held a fascination for over 25 years. A new upper bound of 26 is produced. More important, a new methodology is described for finding upper bounds. The novelty is two-fold. First, parallel disks are employed. This allows 1.4 × 1012 states representing symmetrized cosets to be enumerated in seven terabytes. Second, a fast...

In this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group SO−(2n, 2r). Then we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the...

Let G be a real form of a complex reductive group. Suppose that we are given involutions σ and θ of G. Let H = G denote the fixed group of σ and let K = G denote the fixed group of θ. We are interested in calculating the double coset space H\G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our res...