J.H. van Lint

Equilateral point sets in elliptic geometry

This chapter highlights equilateral point sets in elliptic geometry. Elliptic space of r−1 dimensions E r−1 is obtained from r -dimensional vector space R r with inner product ( a , b ). For 1 , any k -dimensional linear subspace R k of R r is called a ( k−1 )-dimensional elliptic subspace E k−1 . The query for equilateral point sets in elliptic geometry leads to the search for matrices B of order n and elements whose smallest eigenvalue has a high multiplicity. For n elliptic points A 1 , A 2 , …, A n , carried by the unit vectors a 1 , …, a n and spanning elliptic space E r−1 , the Gram matrix is symmetric, semipositive definite, and of rank r . B -matrices of order n ≡ 2r that have only two distinct eigenvalues with equal multiplicities r are called C -matrices. In view of the existence of a Hadamard matrix of order 92, it is interesting to know whether Paleys construction may be reversed to obtain a C -matrix of order 46.

On the minimum distance of cyclic codes

The main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds (i.e., BCH bound, HT bound, Roos bound). This bound is related to a second method for bounding the minimum distance of a cyclic code, which we call shifting. This method can be even stronger than the first one. For all binary cyclic codes of length (with two exceptions), we show that our methods yield the true minimum distance. The two exceptions at the end of our list are a code and its even-weight subcode. We treat several examples of cyclic codes of length \geq 63 .

Repeated-root cyclic codes

In the theory of cyclic codes, it is common practice to require that (n,q)=1, where n is the word length and F/sub q/ is the alphabet. It is shown that the even weight subcodes of the shortened binary Hamming codes form a sequence of repeated-root cyclic codes that are optimal. In nearly all other cases, one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime. This statement is based on an analysis for lengths up to 100. A theorem shows why this was to be expected, but it also leads to low-complexity decoding methods. This is an advantage, especially for the codes that are not much worse than corresponding codes of odd length. It is demonstrated that a binary cyclic code of length 2n (n odd) can be obtained from two cyclic codes of length n by the well-known mod u mod u+v mod construction. This leads to an infinite sequence of optimal cyclic codes with distance 4. Furthermore, it is shown that low-complexity decoding methods can be used for these codes. The structure theorem generalizes to other characteristics and to other lengths. Some comparisons of the methods using earlier examples are given. >

Introduction to Coding Theory and Algebraic Geometry

I: Coding Theory.- 1. Finite fields.- 2. Error-correcting codes.- 3. Linear codes.- 4. Cyclic codes.- 5. Classical Goppa codes.- 6. Bounds on codes.- 7. Self-dual codes.- 8. Codes from curves.- References.- II: Algebraic Geometry.- I. Elementary concepts from algebraic geometry.- II. Divisors on algebraic curves.- III. Goppa Codes.- IV. Couting points on curves over finite fields.- V. Shimura curves and codes.- Index of notations.

A survey of perfect codes

• A submitted manuscript is the authors version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publishers website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

Generalized Reed - Solomon codes from algebraic geometry

A few years ago Tsfasman {\em et al.,} using results from algebraic geometry, showed that there is a sequence of codes which are generalizations of Goppa codes and which exceed the Gilbert-Varshamov bound. We show that a similar sequence of codes (in fact, the duals of the previous codes) can be found by generalizing the construction of Reed-Solomon codes. Our approach has the advantage that it uses less complicated concepts from algebraic geometry.

On the Preparata and Goethals codes

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publishers website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

Generalized quadratic residue codes

A simple definition of generalized quadratic residue codes, that is, quadratic residue codes of block length p^{m} , is given, and an account of many of their properties is presented.

The football pool problem for 5 matches

We consider the set N of all 5-tuples x = (xl, x2, x3, x4, xs) with x~ = 0, 1, or 2 for i = 1,..., 5. The problem treated in this paper is determining the minimal k for which a set ~ of 5-tuples exists such that for each x in N there is an element in ~ that differs from x in at most one coordinate.

Genetic algorithms in coding theory: a table for A 3 ( n,d )

Abstract We consider the problem of finding values of A3(n,d), i.e., the maximal size of a ternary code of length n and minimum distance d. Our approach is based on a search for good lower bounds and a comparison of these bounds with known upper bounds. Several lower bounds are obtained using a genetic local search algorithm. Other lower bounds are obtained by constructing codes. For those cases in which lower and upper bounds coincide, this yields exact values of A3(n,d). A table is included containing the known values of the upper and lower bounds for A3(n,d), with n≤16. For some values of n and d the corresponding codes are given.