Lloyd R. Welch

Lower bounds on the maximum cross correlation of signals (Corresp.)

Some communication systems require sets of signals with impulse-like autocorrelation functions and small cross correlation. There is considerable literature on signals with impulse-like autocorrelation functions hut little on sets of signals with small cross correlation. A possible reason is that designers put too severe a restriction on cross correlation magnitudes. This correspondence establishes lower bounds on how small the cross correlation and autocorrelation can simultaneously be.

New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities

With the Delsarte-MacWilliams inequalities as a starting point, an upper bound is obtained on the rate of a binary code as a function of its minimum distance. This upper bound is asymptotically less than Levenshteins bound, and so also Eliass.

Weight distributions of the cosets of the (32,6) Reed-Muller code

In this paper we present the weight distribution of all 2^26 cosets of the (32,6) first-order Reed-Muller code. The code is invariant under the complete affine group, of order 32 \times 31 \times 30 \times 28 \times 24 \times 16. In the Appendix we show (by hand computations) that this group partitions the 2^26 cosets into only 48 equivalence classes, and we obtain the number of cosets in each class. A simple computer program then enumerated the weights of the 32 vectors ih each of the 48 cosets. These coset enumerations also answer this equivalent problem: how well are the 2^32 Boolean functions of five variables approximated by the 2^5 linear functions and their complements?

Continued fractions and Berlekamp's algorithm

Theorems are presented concerning the optimality of rational approximations using non-Archimedean norms. The algorithm for developing the rational approximations is based on continued fraction techniques and is virtually equivalent to an algorithm employed by Berlekamp for decoding BCH codes. Several variations of the continued fraction technique and Berlekamps algorithm are illustrated on a common example.

Group characters: Sequences with good correlation properties

The structure of the group of integers relatively prime to n under multiplication modulo n is reviewed, and the basic properties of characters defined on that group is developed. Appropriately chosen subcollections of the characters when viewed as periodic sequences are then shown to have relatively ideal autocorrelation and cross correlation properties. The results of a computer study indicate that the same subcollections when viewed as finite length sequences also have very good aperiodic autocorrelation and cross correlation properties.

The fast decoding of Reed-Solomon codes using Fermat theoretic transforms and continued fractions

It is shown that Reed-Solomon (RS) codes can be decoded by using a fast Fourier transform (FFT) algorithm over finite fields GF(F_{n}) , where F_{n} is a Fermat prime, and continued fractions. This new transform decoding method is simpler than the standard method for RS codes. The computing time of this new decoding algorithm in software can be faster than the standard decoding method for RS codes.

The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

It is shown that \sqrt\[8]{2} is an element of order 2^{n+4} in GF(F_{n}) , where F_{n}=2^{2^{n}}+1 is a Fermat prime for n=3,4 . Hence it can be used to define a fast Fourier transform (FFT) of as many as 2^{n+4} symbols in GF(F_{n}) . Since \sqrt[8]{2} is a root of unity of order 2^{n+4} in GF(F_{n}) , this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the case n=2 , where \sqrt{2} is in GF(F_{2})=GF(17) , is given.

Mechanization of codes with bounded synchronization delays

Bounded synchronization delay codes have the property that no proper cyclic rearrangement of the letters of a codeword is another codeword. Because of this property research in code design has centered on criteria for selecting one word from each nonperiodic cyclic equivalence class to satisfy various additional constraints. This is all that is necessary when it is feasible for the encoder and decoder to use table look-up procedures. However, even for moderate word length, the dictionary size can be quite large and prove a major obstacle in practical applications. This paper describes a systematic procedure for mapping data sequences into nonperiodic cyclic equivalence classes and for performing the inverse mapping. The scheme is arithmetic in nature and does not require large tables.

Erasure decoding in burst-error channels

Burst-error channels have been used to model a large class of modern communication media, and the problem of communicating reliably through such media has received much study [1]-[9]. Existing techniques include two-way communication schemes that involve error detection and retransmission, and schemes that utilize error correcting codes in code interleaving. The error-detection and retransmission scheme is simple, but its applicability has been restricted to limited environments. On the other hand, the concept of code interleaving has proved to be versatile and effective. Code interleaving distributes the error detection and correction burden among the component codes and thus lowers the overall redundancy requirement. However, the memory characteristics of the burst-error channel have not been used. This omission has prompted the investigation presented in this paper to utilize the inherent information embedded in the code interleaving scheme when used with burst-error channels. The concept of erasure decoding is introduced, leading to some useful coding and decoding strategies. Theoretical formulations are devised to predict code performance, and their validity is verified with computer simulations.

A transform decoder for Reed-Solomon codes in multiple-user communication systems

Encoding and decoding algorithms for Reed-Solomon codes based on Fourier-like transforms on finite field and finite rings are discussed. Classes of codes are proposed for two different types of multiple-user communication systems: a multichannel communication system and a multiaccess communication system. For the first system, a fast decoding algorithm is developed that uses transforms on a finite ring which is isomorphic to a direct sum of Galois fields. For the second system, an efficient (in terms of information rate) coding scheme is proposed which utilizes a direct sum of Galois fields.