Classification of error locator polynomials for double errorcorrecting BCH codes

We give a complete classification of the error locator polynomials that occur in the Berlekamp decoding of double error correcting (DEC) Bose-Chaudhuri-Hocquenghem (BCH) codes. We present a new construction showing that all quadratic error locator polynomials produced by received vectors falling in the interstitial region between decoding spheres are illegitimate and have no roots. Furthermore, we show that a small subset of received vectors in the interstitial region produce cubic error locator polynomials that are illegitimate except for the correctable case of a triple error pattern with three equally spaced errors in the cyclic sense     

Minimum weight convolutional codewords of finite length (Corresp.)

For convolutional codes, thc variation of the minimum distance between nonmerging codewords with the lengths of those codewords is considered for all finite lengths. This is carried out in terms of a new distance parameter for convolutional codes do, the minimum average weight per branch over all cycles. Upper and lower bounds on do for binary convolutional codes of rate1/nare presented. The tradeoff betweend_{o}and the free distanced_{free}is obtained for small memory length codes.     

Binary GCD algorithm for computing error locator polynomials in Reed-Solomon decoding

The binary GCD algorithm, discovered by Stein, is an alternative to the Euclidean algorithm for computing the greatest common divisor of two integers. In this work, the binary GCD algorithm is applied to Reed-Solomon decoding and a novel iterative algorithm for computing error locator polynomials is proposed. Compared to Euclidean-based algorithms, this algorithm exhibits some speed and area advantages.     

Cyclic codes over Z4, locator polynomials, and Newton'sidentities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z₄ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z ₄. Linear codes with length 2^m (m, odd) and size 2(2^m+1-5m-2). The Gray image of the code of length 32 is the best (64, 2³⁷) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z₄ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2³²) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice     

On the problem of finding zero-concurring codewords

Zero-concurring codewords disclose a certain structure of the code that may be used for efficient soft-decision decoding and for designing DC-free codes. Methods for constructing sets of zero-concurring codewords are presented for several families of codes. For the general case an algorithm solution of the problem is offered. A table of results obtained using the proposed techniques is supplied for all the primitive narrow-sense binary BCH codes of length up to 127     

Simple method to increase the minimum weight of turbo codes

A method to increase the minimum free distance of a turbo code is presented by excluding the codewords with minimum and/or sub-minimum weight from the code set. This method needs almost no change in the original encoding and decoding schemes and the improvement is noticeable.     

Minimum-weight codewords as generators of generalized Reed-Mullercodes

We establish the range of values of ρ, where 0⩽ρ⩽m(q-1), for which the generalized Reed-Muller code R_Fq(ρ, m) of length q^m over the field F_q of order q is spanned by its minimum-weight vectors     

Closed solution of Berlekamp's algorithm for fast decoding of BCHcodes

A decision tree solution is presented for the most complicated step in decoding binary BCH codes, namely, the computation of an error location polynomial over GF(2^m) from the syndrome vector of received data. The author runs S. Lin's (1970) iterative version of the Berlekamp-Massey algorithm symbolically, keeping the results at each level in the form of branches of a binary decision tree. A decoder can then be constructed that uses the derived formulas to evaluate a decision variable at each level. Complete traversal of the tree using the decision variables leads to the correct polynomial coefficients for the received vector. The decoder can be implemented in a very straightforward way with a simple processor or program that performs extension field arithmetic, or it can be realized entirely in hardware using lookup tables for multiplications, inverses, and exponents, and exclusive OR operations for addition     

A square root bound on the minimum weight in quasi-cyclic codes

We establish a square root bound on the minimum weight in the quasi-cyclic binary codes constructed by Bhargava, Tavares, and Shiva. The proof rests on viewing the codes as ideals in a group algebra over GF (4). Theorem 6 answers a question raised by F. J. MacWilliams and N. J. A. Sloane in {em The Theory of Error-Correcting Codes.} Theorems 3, 4, and 5 provide information about the way the nonzero entries of a codeword of minimum weight are distributed among the coordinate positions.     

Octet error probability analysis of ADSL–DMT codewords

This paper presents an analytical investigation of the octet error probability of ADSL codewords under the impact of white Gaussian noise. It is shown that the assumption of independent bit errors which can be found in the literature is invalid in this case. Instead, as a novel method, bitgroups from DMT subcarriers contributing to a particular octet are analyzed. Typical bitgroup types are discussed with regard to their error probability. Finally, proposals for applying these results in optimized system solutions are derived.     

Weight distributions of cosets of two-error-correcting binary BCHcodes, extended or not

Let B be the binary two-error-correcting BCH code of length 2^m-1 and let Bˆ be the extended code of B. We give formal expressions of weight distributions of the cosets of the codes Bˆ only depending on m. We can then deduce the weight distributions of the cosets of B. When m is odd, it is well known that there are four distinct weight distributions for the cosets of B. So our main result is about the even case. Camion, Courteau, and Montpetit (see ibid., vol.38, no.7, p.1353, 1992) observe that for the lengths 15, 63, and 255 there are eight distinct weight distributions. We prove that this property holds for the codes Bˆ and B for all even m     

Some lower bounds on the minimum weight of cyclic codes of composite length

An extension of Goethals' results [3] is presented. Cyclic codes of composite block lengthn = n_{1}n_{2}withn_{1}andn_{2}relatively prime are considered. By using a modified form of the Mattson-Solomon formulation [5], some lower bounds on the minimum weight are obtained. These lower bounds improve on the BCH bound [4] for a considerable number of nonprimitive BCH codes.     

Some results on the minimum weight of primitive BCH codes (Corresp.)

For1 leq i leq m - s- 2and0 leq s leq m -2i, the intersection of the binary BCH code of designed distance2 ^{m-s-1} - 2 ^{m-s-t-1} - 1and length2^m - 1with the shortened(s + 2)th-order Reed-Muller code of length2^m -- 1has codewords of weight2^{m-s-1} - 2^{m-s-t-1} - 1.     

A proof of the odd-symmetry of the phases for minimum weight perturbation phase-only null synthesis

It is proved that if phase-only weight control is used to impose nulls in a real pattern of a linear array of isotropic equispaced elements while minimizing the sum of the squares of the absolute weight perturbations then the phase perturbations are odd-symmetric with respect to a phase reference at the array center. Equivalently, the perturbed pattern is real.     

光纤断点测试仪的研制

为解决传统OTDR价格昂贵和操作复杂的问题,研制出了一台初步达到市场商品化标准的光纤断点测试仪.阐述了光纤断点测试仪的工作原理和制作方法,对其每个组成部分,包括光器件、电子器件以及数据采集都做了说明.通过对光纤样品的测量,分析了该光纤断点测试仪的功能与测量效果.     

Optimal binary one-error-correcting codes of length 10 have 72codewords

The maximum number of codewords in a binary code with length n and minimum distance d is denoted by A(n, d). By construction it is known that A(10, 3)⩾72 and A(11, 3)⩾144. These bounds have long been conjectured to be the exact values. This is here proved by classifying various codes of smaller length and lengthening these using backtracking and isomorphism rejection. There are 562 inequivalent codes attaining A(10, 3)=72 and 7398 inequivalent codes attaining A(11, 3)=144     

Construction and characteristics of codewords and pulse shapes that satisfy spectral constraints

Constrained sequence codes and pulse shapes are used to control signal characteristics in digital communication systems. This letter outlines a technique to construct codewords and pulse shapes to meet specified spectral constraints. Based on a block-coded representation of the signal, it is shown how symbol values can be evaluated to satisfy the most significant terms of a Fourier series representation of the desired spectral response. Several properties of the resulting signals are given, and examples are presented to confirm the validity of this approach.     

Do optimal entropy-constrained quantizers have a finite or infinite number of codewords?

An entropy-constrained quantizer Q is optimal if it minimizes the expected distortion D(Q) subject to a constraint on the output entropy H(Q). We use the Lagrangian formulation to show the existence and study the structure of optimal entropy-constrained quantizers that achieve a point on the lower convex hull of the operational distortion-rate function D/sub h/(R) = inf/sub Q/{D(Q) : H(Q) /spl les/ R}. In general, an optimal entropy-constrained quantizer may have a countably infinite number of codewords. Our main results show that if the tail of the source distribution is sufficiently light (resp., heavy) with respect to the distortion measure, the Lagrangian-optimal entropy-constrained quantizer has a finite (resp., infinite) number of codewords. In particular, for the squared error distortion measure, if the tail of the source distribution is lighter than the tail of a Gaussian distribution, then the Lagrangian-optimal quantizer has only a finite number of codewords, while if the tail is heavier than that of the Gaussian, the Lagrangian-optimal quantizer has an infinite number of codewords.     

The study of atmospheric turbulence with an acoustic locator

As a rule, sodar investigations of turbulent atmosphere are based on measurements of the scattered cross section σ. Measurements of the spectrum broadening of the scattered signal provides an additional information. The authors compare scattering cross section data with spectrum broadening. The atmosphere sodar sounding was carried out in Zimenki, 25 km east of Nizhny Novgorod. The bistatic sodar with narrow (3°-5°) antenna beams has allowed the authors to carry out simultaneous measurements of scattering cross section a of acoustic signals with a wavelength λ≃0.5 m and frequency broadening of the scattered signal OF. Results have shown an essential correlation (up to 0.85) between σ and ΔF²