The fast decoding algorithm for one point algebraic-geometry codes of Sakata, Elbrond Jensen, and Hoholdt (see ibid., vol. 41, p. 1762-8, Nov. 1995) corrects all error patterns of weight less than half the Feng-Rao minimum distance. In this correspondence we analyze the performance of the algorithm for heavier error patterns. It turns out that in the typical case, where the error points are "independent", one can prove that the algorithm always fails, that is gives a wrong or no answer, except for high rates where it does much better than expected. This explains the simulation results presented by O'Sullivan at the IEEE Int. Symp. Information Theory, Ulm, Germany (1997). We also show that for dependent errors the algorithm almost always corrects these.

In the decoding of one-point algebraic-geometry codes with one defining equation in two variables we generalize the Forney formula for the Bose-Chaudhuri-Hocqungham (BCH) codes.

This article gives an errata (that is erasure- and error-) decoding algorithm of one-point algebraic-geometry codes up to the Feng-Rao (1994) designed minimum distance using Sakata's (see Proc. 1995 IEEE Int. Symp. Information Theory, Whistler, BC, Canada, 1995) multidimensional generalization of the Berlekamp-Massey (1969) algorithm and the voting procedure of Feng and Rao.

We give necessary and sufficient conditions for a binary linear code to be Z/sub 4/-linear. Especially we treat optimal, binary linear codes and determine all such codes with minimum weight less or equal to six which are Z/sub 4/-linear.

We treat a general class of algebraic-geometric codes and show how to decode these up to half the Feng-Rao bound, using an extension and modification of the Sakata algorithm (1990). The Sakata algorithm is a generalization to N dimensions of the classical Berlekamp-Massey algorithm.

We present a decoding algorithm for algebraic-geometric codes from regular plane curves, in particular the Hermitian curve, which corrects all error patterns of weight less than d*/2 with low complexity. The algorithm is based on the majority scheme of Feng and Rao (1993) and uses a modified version of Sakata's (1988) generalization of the Berlekamp-Massey algorithm.

An algorithm that, for codes from a regular plane curve, corrects up to (d*/2)-(m/sup 2//8)+(m/4)-(9/8) errors, where d* is the designed distance and m is the degree of the curve, was presented in an earlier work (see ibid., vol.35, p.811-21, 1989). It is now shown that this bound is the best possible for the algorithm considered.>

Improvement to an earlier decoding algorithm for codes from algebraic geometry is presented. For codes from an arbitrary regular plane curve the authors correct up to d*/2-m/sup 2//8+m/4-9/8 errors, where d* is the designed distance of the code and m is the degree of the curve. The complexity of finding the error locator is O(n/sup 7/3/), where n is the length of the code. For codes from Hermitian curves the complexity of finding the error values, given the error locator, is O(n/sup 2/), and the same complexity can be obtained in the general case if only d*/2-m/sup 2//2 errors are corrected.>

Long binary sequences related to cyclic difference sets are investigated. Among all known constructions of cyclic difference sets it is shown that only sequences constructed from Hadamard difference sets can have an asymptotic nonzero merit factor. Maximal-length shift register sequences, Legendre, and twin-prime sequences are all constructed from Hadamard difference sets. The authors prove that the asymptotic merit factor of any maximal-length shift register sequence is three. For twin-prime sequences it is shown that the best asymptotic merit factor is six. This value is obtained by shifting the twin-prime sequence one quarter of its length. It turns out that Legendre sequences and twin-prime sequences have similar behavior. Jacobi sequences are investigated on the basis of the Jacobi symbol. The best asymptotic merit factor is shown to be six. Through the introduction of product sequences, it is argued that the maximal merit factor among all sequences of length N is at least six when N is large. The authors also demonstrate that it is fairly easy to construct sequences of moderate composite length with a merit factor close to six.>

A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result is a decoding algorithm which turns out to be a generalization of the Peterson algorithm for decoding BCH decoder codes.>

M.J.E. Golay (ibid., vol.IT-23, no.1, p.43-51, 1977) has used the ergodicity postulate to calculate that the merit factor F of a Legendre sequence offset by a fraction f of its length has an asymptotic value given by 1/F=(2/3)-4 mod f mod +8f/sup 2/, mod f mod>

Series representations of the form f(t) approximately Sigma /sub n=- infinity //sup infinity / Sigma /sub k=- infinity //sup infinity /a/sub n,k/ nu (t-n)/sub e//sup 2 pi /kts/ for bounded signals f(t) are studied, as are conditions on the unit function nu (t), such that coefficients a/sub n,k/ reveal the energy content of f(t) in the time interval n-(1/2)>

A class of infinite\pm 1sequences is presented whose autocorrelation function is zero for all nonzero shifts.

A class of binary sequences of lengthN = 2^{m}is considered, and it is shown that their aperiodic autocorrelations can be calculated recursively in a simple way. Based on this, the merit factor of the sequences is calculated and it is shown that the asymptotic value is3. Finally, it is proved that the magnitude of the maximal aperiodic autocorrelation is bounded byN^{0.9}.