Tom Høholdt

Algebraic-geometry codes

The theory of error-correcting codes derived from curves in an algebraic geometry was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. The development of the theory has received intense consideration since that time and the purpose of the paper is to review this work. Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced. Code constructions from particular classes of curves, including the Klein quartic, elliptic, and hyperelliptic curves, and Hermitian curves, are presented. Decoding algorithms for these classes of codes, and others, are considered. The construction of classes of asymptotically good codes using modular curves is also discussed.

Construction and decoding of a class of algebraic geometry codes

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. >

Fast decoding of codes from algebraic plane curves

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. >

Determination of the merit factor of Legendre sequences

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 >

Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound

Summary form only given, as follows. Efficient decoding of BCH- and Reed-Solomon codes can be done using the Berlekanp-Massey (1969) algorithm, and it is natural to try to use the extension of this to N dimensions of Sakata (see Inform. Computat., vol.84, no.2, p.207, 1990) to decode algebraic geometry codes. We treat a general class of algebraic geometry codes and show how to decode these up to half the Feng-Rao (see IEEE Trans. Inform. Theory, vol.IT 39, no.1 p.37-45, 1993) bound, using an extension and modification of the Sakata algorithm. >

The merit factor of binary sequences related to difference sets

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. >

Ternary sequences with perfect periodic autocorrelation (Corresp.)

We construct 0, \pm 1 sequences of length (q^{2l+1}-l)/(q-1) , where q=2^{s} , with out-of-phase periodic autocorrelation 0 , and in-phase correlation q^{2l}; SUch that the peak factor of radiation is (q^{2l+1}-1) /(q^{2+l}-q^{2l}) , which is close to 1 as q becomes large.

Fast decoding of algebraic-geometric codes up to the designed minimum distance

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 Sakatas (1988) generalization of the Berlekamp-Massey algorithm.

Decoding Hermitian Codes with Sudan's Algorithm

We present an efficient implementation of Sudans algorithm for list decoding Hermitian codes beyond half the minimum distance. The main ingredients are an explicit method to calculate so-called increasing zero bases, an efficient interpolation algorithm for finding the Q- polynomial, and a reduction of the problemof factoring the Q-polynomial to the problem of factoring a univariate polynomial over a large finite field.

Bounds on list decoding of MDS codes

We derive upper bounds on the number of errors that can be corrected by list decoding of maximum-distance separable (MDS) codes using small lists. We show that the performance of Reed-Solomon (RS) codes, for certain parameter values, is limited by worst case codeword configurations, but that with randomly chosen codes over large alphabets, more errors can be corrected.