Lattice constellations and codes from quadratic number fields

We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric modulo a two-dimensional (2-D) grid. In particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate.

Integrating open source codes for solid modeling and automatic mesh generation in FEM applications

The applications of the Finite Element Method (FEM) for three-dimensional domains are already well documented in the framework of Computational Electromagnetics. However, despite the power and reliability of this technique for solving partial differential equations, there are only a few examples of open source codes available and dedicated to the solid modeling and automatic constrained tetrahedralization, which are the most time consuming steps in a typical three-dimensional FEM simulation. Besides, these open source codes are usually developed separately by distinct software teams, and even under conflicting specifications. In this paper, we describe an experiment of open source code integration for solid modeling and automatic mesh generation. The integration strategy and techniques are discussed, and examples and performance results are given, specially for complicated and irregular volumes which are not simply connected. © 2011 IEEE.

Códigos, reglamentos y normas sobre abastecimiento de agua, saneamiento y eliminación de desechos sólidos con especial atención a las necesidades de las comunidades de bajos ingresos en América Latina y el Caribe = Codes, regulations and standards on water supply, sanitation and solid-waste disposal with emphasis on low-income community requirements in Latin America and the Caribbean

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Commutative group codes in R4, R6, R8 and R16-Approaching the bound

Spherical codes in even dimensions n = 2m generated by a commutative group of orthogonal matrices can be determined by a quotient of m-dimensional lattices when the sublattice has an orthogonal basis. We discuss here the existence of orthogonal sublattices of the lattices A2, D3, D4 and E8, which have the best packing density in their dimensions, in order to generate families of commutative group codes approaching the bound presented in Siqueira and Costa (2008) [14]. © 2013 Elsevier B.V. All rights reserved.

A BCH code and a sequence of cyclic codes

This study establishes that for a given binary BCH code C0 n of length n generated by a polynomial g(x) ∈ F2[x] of degree r there exists a family of binary cyclic codes {Cm 2m−1(n+1)n}m≥1 such that for each m ≥ 1, the binary cyclic code Cm 2m−1(n+1)n has length 2m−1(n + 1)n and is generated by a generalized polynomial g(x 1 2m ) ∈ F2[x, 1 2m Z≥0] of degree 2mr. Furthermore, C0 n is embedded in Cm 2m−1(n+1)n and Cm 2m−1(n+1)n is embedded in Cm+1 2m(n+1)n for each m ≥ 1. By a newly proposed algorithm, codewords of the binary BCH code C0 n can be transmitted with high code rate and decoded by the decoder of any member of the family {Cm 2m−1(n+1)n}m≥1 of binary cyclic codes, having the same code rate.

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.

Goppa codes through polynomials of B[X;(1/2^2)Z_0] and its decoding principle

In this paper, we present a decoding principle for Goppa codes constructed by generalized polynomials, which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight $t\leq 2r$, i.e., whose minimum Hamming distance is $2^{2}r+1$.

Cyclic codes through B[X;(a/b)Z_0, with (a/b) in Q^{+} and b=a+1, and Encoding

Let B[X; S] be a monoid ring with any fixed finite unitary commutative ring B and is the monoid S such that b = a + 1, where a is any positive integer. In this paper we constructed cyclic codes, BCH codes, alternant codes, Goppa codes, Srivastava codes through monoid ring . For a = 1, almost all the results contained in [16] stands as a very particular case of this study.

Packet erasure correcting codes for wireless sensor networks: implementation and field trial measurements

This thesis regards the Wireless Sensor Network (WSN), as one of the most important technologies for the twenty-first century and the implementation of different packet correcting erasure codes to cope with the ”bursty” nature of the transmission channel and the possibility of packet losses during the transmission. The limited battery capacity of each sensor node makes the minimization of the power consumption one of the primary concerns in WSN. Considering also the fact that in each sensor node the communication is considerably more expensive than computation, this motivates the core idea to invest computation within the network whenever possible to safe on communication costs. The goal of the research was to evaluate a parameter, for example the Packet Erasure Ratio (PER), that permit to verify the functionality and the behavior of the created network, validate the theoretical expectations and evaluate the convenience of introducing the recovery packet techniques using different types of packet erasure codes in different types of networks. Thus, considering all the constrains of energy consumption in WSN, the topic of this thesis is to try to minimize it by introducing encoding/decoding algorithms in the transmission chain in order to prevent the retransmission of the erased packets through the Packet Erasure Channel and save the energy used for each retransmitted packet. In this way it is possible extend the lifetime of entire network.