Sequences of Primitive and Non-primitive BCH Codes

Abstract

In this work, we introduce a method by which it is established that; how a sequence of non-primitive BCH codes can be obtained by a given primitive BCH code. For this, we rush to the out of routine assembling technique of BCH codes and use the structure of monoid rings instead of polynomial rings. Accordingly, it is gotten that there is a sequence $\{C_{b^{j}n}\}_{1\leq j\leq m}$, where $b^{j}n$ is the length of $C_{b^{j}n}$, of non-primitive binary BCH codes against a given binary BCH code $C_{n}$ of length $n$. Matlab based simulated algorithms for encoding and decoding for these type of codes are introduced. Matlab provides built in routines for construction of a primitive BCH code, but impose several constraints, like degree $s$ of primitive irreducible polynomial should be less than $16$. This work focuses on non-primitive irreducible polynomials having degree $bs$, which go far more than 16.

Rotated Z^n-Lattices via Real Subfields of Q(\zeta_2r)

“…Let L be the cyclotomic field Q(ζ 2 3 ) and K its maximal real subfield Q(ζ 2 3 + ζ −1 2 3 ). In this case, α = 2 − (ζ 3 2 3 + ζ −3 2 3 ), ∆ K = 2 3 and c = 2 2 . Considering the Z-basis {e 0 = 1, e 1 = e 1 = ζ 2 3 + ζ −1 2 3 } for O K and Q(x, y) = 1 2 2 q α (x, y) = 1 2 2 Tr K/Q (αxy), it follows that the matrix of q α is given by…”

“…Ring theory and algebric number theory have long shown to be useful tools in the theory of information and coding [8] and [2]. In particular, lattices (discrete subgroups of the Euclidean n-space R n ) have played a relevant role in code design for different types of channels, see for example [12], [5], [6], and [9].…”

“…In [1], for any integer r ≥ 3, rotated Z n -lattices, n = 2 r−2 , were constructed from Q(ζ 2 r ) + = Q(ζ 2 r + ζ −1 2 r ), the maximal real subfield of Q(ζ 2 r ), where ζ 2 r is a primitive 2 r -th root of unity. In this work, we extend the method in [1] by considering a particular subfield of Q(ζ 2 r ) + , namely, Q(ζ 2 2 r + ζ −2 2 r ), to construct lattices in dimensions that are powers of 2. This paper is organized as follows: In Section 2, notions and results from algebraic number theory that are used in the work are reviewed.…”

Constructions of Dense Lattices of Full Diversity

“…If M = {a 1 (ζ 9 + ζ −1 9 ) + a 2 (ζ 2 9 + ζ −2 9 ) + a 3 (ζ 3 9 + ζ −3 9 ) ∈ O K : 4a 1 + 4a 2 + a 3 ≡ 0(mod 6) and a 3 ≡ 0(mod 2)}, then M is a Z-submodule of O K of rank 3 and index 6. From Theorem 6, the trace form of α ∈ M is given by Tr K/Q (α 2 ) = 18(a 2 1 + a 1 a 2 + 4a 1 a 3 + a 2 2 + 4a 2 a 3 + 2a 2 3 ).…”

“…Lattices are discrete subgroups of Euclidean n-space, R n , and they have been considered in different applied areas, especially in coding/modulation theory and more recently in cryptography. Algebraic lattices are those obtained via number fields and they have been studied in several papers and from different points of view, see [1,2,3,5,6,10]. These algebraic lattices are constructed through the canonical homomorphism via Z-modules of the ring of algebraic integers of a number field.…”