Alternant and BCH codes over certain rings

Alternant codes over arbitrary finite commutative local rings with identity are constructed in terms of parity-check matrices. The derivation is based on the factorization of x s - 1 over the unit group of an appropriate extension of the finite ring. An efficient decoding procedure which makes use of the modified Berlekamp-Massey algorithm to correct errors and erasures is presented. Furthermore, we address the construction of BCH codes over Zm under Lee metric.

Construction and decoding of BCH codes over finite commutative rings

BCH codes over arbitrary finite commutative rings with identity are derived in terms of their locator vector. The derivation is based on the factorization of xs -1 over the unit ring of an appropriate extension of the finite ring. We present an efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, for these codes. The code construction and the decoding procedures are very similar to the BCH codes over finite integer rings. © 1999 Elsevier B.V. All rights reserved.

Resolução de equações algébricas por radicais

Pós-graduação em Educação Matemática - IGCE

Forma traço sobre algumas extensões galoisianas de corpos p-Ádicos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sobre revoluções científicas na matemática

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Uma análise da parte primeira da obra Sulla risoluzione delle equazioni algebriche, de Enrico Betti

Pós-graduação em Educação Matemática - IGCE

Construções euclidianas e o desfecho de problemas famosos da geometria

Construction techniques with ruler and the compasses, fundamental on Euclidean geometry, have been related to modern algebraic theories such as solving equations and extension of bodies from the works by Paolo Ruffini (1765-1822), Niels Henrik Abel (1802-1829) and Evariste Galois (1811-1832). This relation could provide an answer to some famous problems, from ancient Greece, such as doubling the cube, the trisection Angle, the Quadrature of the Circle and the construction of regular polygons, which remained unsolved for over two thousand years. Also important for our purposes are the notions of algebraic numbers, transcendental and the criteria for constructability, of those numbers. The objective of this study is to reconstruct relevant steps of geometric constructions with ruler (unmarked) and the compasses, from the elementary to the outcome buildings, in the nineteenth century, considering those mentioned problems.

Construction and decoding of BCH codes over chain of commutative rings

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A0 ⊂ A1 ⊂···⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K0 ⊂ K1 ⊂···⊂ Kt−1 ⊂ Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, A∗ i and K∗ i are the groups of units of Ai and Ki, respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A∗ i and K∗ i for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

Uma construção de códigos BCH

Um código BCH C (respectivamente, um código BCH C 0 ) de comprimento n sobre o anel local Zp k (respectivamente, sobre o corpo Zp) é um ideal no anel Zpk [X] (Xn−1) (respectivamente, no anel Zp[X] (Xn−1) ), que ´e gerado por um polinômio mônico que divide Xn−1. Shankar [1] mostrou que as raízes de Xn−1 são as unidades do anel de Galois GR(p k , s) (respectivamente, corpo de Galois GF(p, s)) que é uma extensão do anel Zp k (respectivamente, do corpo Zp), onde s é o grau de um polinômio irredutível f(X) ∈ Zp k [X]. Neste estudo, assumimos que para si = b i , onde b é um primo e i é um inteiro não negativo tal que 0 ≤ i ≤ t, existem extensões de anéis de Galois correspondentes GR(p k , si) (respectivamente, extensões do corpo de Galois GF(p, si)) do anel Zp k (respectivamente, do corpo Zp). Assim, si = b i para i = 2 ou si = b i para i > 2. De modo análogo a [1], neste trabalho, apresentamos uma sequência de códigos BCH C0, C1, · · · , Ct−1C sobre Zp k de comprimentos n0, n1, · · · , nt−1, nt , e uma sequência de códigos BCH C 0 0 , C0 1 , · · · , C0 t−1 , C0 sobre Zp de comprimentos n0, n1, · · · , nt−1, nt , onde cada ni divide p si − 1. Palavras Chave: Anel de Galois, corpo de Galois, código BCH.

A note on units of finite local rings in an ascending chain

In this paper we present matrices over unitary finite commutative local rings connected through an ascending chain of containments, whose elements are units of the corresponding rings in the chain such that the McCoy ranks are the largest ones.

Linear codes over finite local rings in a chain

For a positive integer $t$, let \begin{equation*} \begin{array}{ccccccccc} (\mathcal{A}_{0},\mathcal{M}_{0}) & \subseteq & (\mathcal{A}_{1},\mathcal{M}_{1}) & \subseteq & & \subseteq & (\mathcal{A}_{t-1},\mathcal{M}_{t-1}) & \subseteq & (\mathcal{A},\mathcal{M}) \\ \cap & & \cap & & & & \cap & & \cap \\ (\mathcal{R}_{0},\mathcal{M}_{0}^{2}) & & (\mathcal{R}_{1},\mathcal{M}_{1}^{2}) & & & & (\mathcal{R}_{t-1},\mathcal{M}_{t-1}^{2}) & & (\mathcal{R},\mathcal{M}^{2}) \end{array} \end{equation*} be a chain of unitary local commutative rings $(\mathcal{A}_{i},\mathcal{M}_{i})$ with their corresponding Galois ring extensions $(\mathcal{R}_{i},\mathcal{M}_{i}^{2})$, for $i=0,1,\cdots,t$. In this paper, we have given a construction technique of the cyclic, BCH, alternant, Goppa and Srivastava codes over these rings. Though, initially in \cite{AP} it is for local ring $(\mathcal{A},\mathcal{M})$, in this paper, this new approach have given a choice in selection of most suitable code in error corrections and code rate perspectives.

Bit-parallel coprocessor for standard ECC-GF(2m) on FPGA

This paper presents the design of a high-speed coprocessor for Elliptic Curve Cryptography over binary Galois Field (ECC- GF(2m)). The purpose of our coprocessor is to accelerate the scalar multiplication performed over elliptic curve points represented by affine coordinates in polynomial basis. Our method consists of using elliptic curve parameters over GF(2163) in accordance with international security requirements to implement a bit-parallel coprocessor on field-programmable gate-array (FPGA). Our coprocessor performs modular inversion by using a process based on the Stein's algorithm. Results are presented and compared to results of other related works. We conclude that our coprocessor is suitable for comparing with any other ECC-hardware proposal, since its speed is comparable to projective coordinate designs.

Fields of two power conductor

The goal of this work is find a description for fields of two power conductor. By the Kronecker-Weber theorem, these amounts to find the subfields of cyclotomic field $\mathbb{Q}(\xi_{2^r})$, where $\xi_{2^r}$ is a $2^r$-th primitive root of unit and $r$ a positive integer. In this case, the cyclotomic extension isn't cyclic, however its Galois group is generated by two elements and the subfield can be expressed by $\mathbb{Q}(\theta)$ for a $\theta\in\mathbb{Q}(\xi_{2^r})$ convenient.