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- E. M. Rains, N.J.A. Sloane
- 1983

A survey of self-dual codes, written for the Handbook of Coding Theory. Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this chapter include codes over F2, F3, F4, Fq, Z4, Zm, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason… (More)

- N.J.A. Sloane
- 2000

This paper gives a brief survey of binary single-deletion-correcting codes. The Varshamov-Tenengolts codes appear to be optimal, but many interesting unsolved problems remain. The connections with shift-register sequences also remain somewhat mysterious.

- N.J.A. Sloane
- 1997

How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, E. M. Rains and P. W. Shor. We have found many nice… (More)

- N.J.A. Sloane
- 1998

A brief report on recent work on the sphere-packing problem. 1991 Mathematics Subject Classification: 52C17

There is a beautiful analogy between most of the notions for lattices and codes and it seems to be quite promising to develop coding theory analogues of concepts known in the theory of lattices and modular forms and vice versa. Some of these analogies are presented in this short note that intends to survey recent developments connected to my talk… (More)

- G. Nebe, E. M. Rains, N.J.A. Sloane
- 2000

A certain family of orthogonal groups (called “Clifford groups” by G. E. Wall) has arisen in a variety of different contexts in recent years. These groups have a simple definition as the automorphism groups of certain generalized Barnes-Wall lattices. This leads to an especially simple construction for the usual Barnes-Wall lattices. {This is based on the… (More)

- V.A. Vaishampayan, N.J.A. Sloane
- 2006 IEEE Information Theory Workshop - ITW '06…
- 2006

The problem of encoding and decoding binary block codes of length n and constant Hamming weight w is formulated as a polytope dissection problem. This is done by working with a w-dimensional Euclidean space representation for the information and code vectors. Novel algorithms based on two new dissections are presented. The first is a dissection of a subset… (More)

- Chao Tian, V.A. Vaishampayan, N.J.A. Sloane
- Proceedings. International Symposium on…
- 2005

We present a novel technique for encoding and decoding constant weight binary codes that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a Euclidean space of dimension equal to the weight of the code. The encoder and decoder mappings are then interpreted as a bijection between a certain hyper-rectangle… (More)

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