Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over ¡ 4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the ¡ 4 domain implies that the binary images have dual weight distributions. The Kerdock and ‘Preparata ’ codes are duals over ¡ 4 — and the Nordstrom-Robinson code is self-dual — which explains why their weight distributions are dual to each other. The Kerdock and ‘Preparata ’ codes are ¡ 4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over ¡ 4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the ‘Preparata ’ code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over ¡ 4, but extended Hamming codes of length n ≥ 32 and the

In this paper, we present a robust and mathematically sound ray-intersection algorithm for implicit surfaces. The algorithm is guar-anteed to numerically find the nearest intersection of the surface with a ray, and is guaranteed not to miss fine features of the surface. It does not require fine tuning or human choice of interactive parameters. Instead, it requires two upper bounds: "L" that limits the net rate of change of the implicit surface function f(x, y, z) and "G" that limits the rate of change of the gradient. We refer to an implicit surface with these rate limits as an "LG-implicit surface." Existing schemes to intersect a ray with an implicit surface have typically been guaranteed to work only for a limited set of implicit functions, such as quadric surfaces or polynomials, or else have been ad-hoc and have not been guaranteed to work. Our technique signifi-cantly extends the ability to intersect rays with implicit surfaces in a guaranteed fashion.

. We present a new algorithm for finding an irreducible polynomial of specified degree over a finite field. Our algorithm is deterministic, and it runs in polynomial time for fields of small characteristic. We in fact prove the stronger result that the problem of finding irreducible polynomials of specified degree over a finite field is deterministic polynomial time reducible to the problem of factoring polynomials over the prime field. 1980 Mathematics Subject Classification (1985 revision). Primary 11T06. This research was supported by National Science Foundation grants DCR-8504485 and DCR-8552596. Appeared in Mathematics of Computation 54, pp. 435--447, 1990. A preliminary version of this paper appeared in Proceedings of the 29th Annual Symposium on Foundations of Computer Science, October 1988. 1. Introduction In this paper we present some new algorithms for finding irreducible polynomials over finite fields. Such polynomials are used to implement arithmetic in extension fields ...

We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. 1.

Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves.

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There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [19-22], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, as well as the papers

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ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.

�������� � In his first two letters to G. H. Hardy and in his notebooks, Ramanujan recorded many theorems about the Rogers–Ramanujan continued fraction. In his lost notebook, he offered several further assertions. The purpose of this paper is to provide proofs for many of the claims about the Rogers–Ramanujan and generalized Rogers–Ramanujan continued fractions found in the lost notebook. These theorems involve, among other things, modular equations, transformations, zeros, and class invariants. 1.