Richard A. Games

A fast algorithm for determining the complexity of a binary sequence with period 2^n (Corresp.)

The complexity of a periodic sequence (s) is defined to be the least number of stages in a linear feedback shift register that generates (s) .

On the complexities of de Bruijn sequences

Abstract The shortest linear recursion which generates a de Bruijn sequence is defined to be the complexity of the sequence. It is shown that the complexity of a binary de Bruijn sequence of span n is bounded by 2 n − 1 from above and 2 n − 1 + n from below. Results on the distribution of the complexities are also presented.

Sonar sequences from Costas arrays and the best known sonar sequences with up to 100 symbols

The authors construct new and improved sonar sequences by applying rotation, multiplication, and shearing transformations to Costas sequence constructions. A catalog of the best known sonar sequences with up to 100 symbols is given. >

The geometry of quadrics and correlations of sequences (Corresp.)

Nondegenerate quadrics of PG (2l, 2^{s}) have been used to construct ternary sequences of length (2^{2sl+1} - 1)/(2^{s} - 1) with perfect autocorrelation function. The same construction can be used for degenerate quadrics for this case as well as quadrics of PG (N,q) , with N arbitrary and q = p^{s} , for any prime p . This is possible because it is shown that if Q \subseteq {\rm PG} (N, q) is a quadric, possibly degenerate, that has the same size as a hyperplane, then, provided Q itself is not a hyperplane, the hyperplanes of PG (N,q) intersect Q in three sizes. These sizes depend on whether N is even or odd and the degeneracy of Q . Finally, a connection to maximum period linear recursive sequences is made.

Optimal book embeddings of the FFT, benes, and barrel shifter networks

This paper gives 3-page book embeddings of three important interconnection networks: the FFT network, the Benes rearrangeable permutation network, and the barrel shifter network. Since all three networks are eventually nonplanar, they require three pages and the present embeddings are optimal. Also, the embeddings have pages of comparable widths.The embeddings of the FFT and barrel shifter networks are obtained by decomposing these recursively defined networks into smaller copies of themselves and using induction. The embedding of the Benes network uses a novel decomposition of the network into eight copies of a smaller FFT network.

On the linear span of binary sequences obtained from q-ary m-sequences, q odd

A class of periodic binary sequences that are obtained from q-ary m-sequences is defined, and a general method to determine their linear spans (the length of the shortest linear recursion over the Galois field GF(2) satisfied by the sequence) is described. The results imply that the binary sequences under consideration have linear spans that are comparable with their periods, which can be made very long. One application of the results shows that the projective and affine hyperplane sequences of odd order both have full linear span. Another application involves the parity sequence of order n, which has period p/sup m/-1, where p is an odd prime. The linear span of a parity sequence of order n is determined in terms of the linear span of a parity sequence of order 1, and this leads to an interesting open problem involving primes. >

A generalized recursive construction for de Bruijn sequences

This paper is concerned with the construction of de Bruijn sequences of span n --binary sequences of period 2^{n} in which every binary n -tuple appears as some n consecutive terms in one period of the sequence. Constructions in the literature are based on maximum length linear sequences, algorithms which start from scratch, and recursive methods which start with a single de Bruijn sequence of span n and produce one of span n+1 . We give a more general recursive construction which takes two de Brnijn sequences of span n and produces a de Bruijn sequence of span n+1 . In addition, for a special case of the construction, the complexity (shortest linear recursion that generates the sequence) of the resulting sequence is determined in terms of the complexities of the ingredient sequences. In particular, de Bruijn sequences of span n+1 with maximum complexity 2^{(n+1)}-1 are obtained from maximum complexity sequences of span n . Reverse-complementary de Bruijn sequences are also considered.

Complex approximations using algebraic integers

The problem of approximating complex numbers by elements of Z[\omega] , the algebraic integers of Q(\omega) , where \omega is a primitive n th root of unity, is considered. The motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete Fourier transform. Smallest error tolerances for the case of eighth roots of unity are derived using a geometric argument. Scale factors involved are reduced from \alpha to \sqrt{\alpha} for this case with roughly the same percentage errors. The case of sixteenth roots of unity gives even better range reductions and is considered only briefly.

(n,k,t))-covering systems and error-trapping decoding (Corresp.)

The technique of error-trapping decoding for algebraic codes is studied in combinatorial terms of covering systems. Let n, k , and t be positive integers such that n \geq k \geq t > 0 . An (n, k,t) -covering system is a pair (X, \beta) , where X is a set of size n and \beta is a collection of subsets of X , each of size k , such that for all T \subseteq X of size t , there exists at least one B \in \beta with T\subseteq B . Let b(n, k, t) denote the smallest size of \beta , such that (X, \beta) is an (n, k, t) -covering system. It is shown that the complexity of an error-trapping decoding technique is bounded by b(n, k, t) from below. Two new methods for constructing small (n, k, t) -covering systems, the algorithmic method and the difference family method, are given.

On the quadratic spans of DeBruijn sequences

The quadratic span of a periodic binary sequence is defined to be the length of the shortest quadratic feedback shift register (FSR) that generates it. An algorithm for computing the quadratic span of a binary sequence is described. The required increase in quadratic span is determined for the special case of when a discrepancy occurs in a linear FSR that generates an initial portion of a sequence. The quadratic spans of binary DeBruijn sequences are investigated. An upper bound for the quadratic span of a DeBruijn sequence of span n is given; this bound is attained by the class of DeBruijn sequences obtained from m-sequences. It is easy to see that a lower bound is n+1, but a lower bound of n+2 is conjectured. The distributions of quadratic spans of DeBruijn sequences of span 3, 4, 5 and 6 are presented. >