Ray Hill

Caps and codes

Abstract The packing problem in the theory of caps is that of finding, or at least bounding, the size m ( r, q ) of an ovaloid (cap of largest size) in the projective space S r,q of dimension r over a field of q elements. This problem and that of constructing and classifying ovaloids are approached by consideration of certain codes associated with caps. Improved general upper bounds on m ( r, q ) are found, which give m (5, 3)⩽56 as a particular case. A 56-cap in S 5,3 is constructed via its code and its uniqueness as an ovaloid is demonstrated.

Optimal ternary linear codes

Let nq(k, d) denote the smallest value of n for which there exists a linear [n, k, d]-code over GF(q). An [n, k, d]-code whose length is equal to nq(k, d) is called optimal. The problem of finding nq(k, d)has received much attention for the case q = 2. We generalize several results to the case of an arbitrary prime power q as well as introducing new results and a detailed methodology to enable the problem to be tackled over any finite field.In particular, we study the problem with q = 3 and determine n3(k, d) for all d when k ≤ 4, and n3(5, d) for all but 30 values of d.

Optimal ternary quasi-cyclic codes

Quasi-cyclic codes have provided a rich source of good linear codes. Previous constructions of quasi-cyclic codes have been confined mainly to codes whose length is a multiple of the dimension. In this paper it is shown how searches may be extended to codes whose length is a multiple of some integer which is greater than the dimension. The particular case of 5-dimensional codes over GF(3) is considered and a number of optimal codes (i.e., [n, k, d]-codes having largest possible minimum distance d for given length n and dimension k) are constructed. These include ternary codes with parameters [45, 5, 28], [36, 5, 22], [42, 5, 26], [48, 5, 30] and [72, 5, 46], all of which improve on the previously best known bounds.

An Extension Theorem for Linear Codes

One of the first results one meets in coding theory is that a binary linear [n,k,d] code, whose minimum distance is odd, can be extended to an [n + 1, k, d + 1] code. This is one of the few elementary results about binary codes which does not obviously generalise to q-ary codes. The aim of this paper is to give a simple sufficient condition for a q-ary [n, k, d] code to be extendable to an [n + 1, k, d + 1] code. Applications will be given to the construction and classification of good codes, to proving the non- existence of certain codes, and also an application in finite geometry.

Extensions of linear codes

One of the first results one meets in coding theory is that a binary linear [n,k,d]-code, whose minimum weight is odd, can be extended to an [n+1,k,d+1]-code. This is one of the few elementary results about binary codes which does not obviously generalize to q-ary codes. Although one can readily extend a q-ary code, by adding a further check digit, it is not clear under what circumstances such an extension will increase the minimum distance. The aim of this paper is to give a simple sufficient condition for a q-ary [n,k,d]-code to be extendable to an [n+1,k,d+1]-code. The result is indeed a generalization of the above result for binary codes. It also generalizes a result for ternary codes due to van Eupen and Lisonek, whose proof made use of quadratic form. The present generalization has an elementary proof.

Optimal linear codes over GF(4)

Abstract A central problem in coding theory is that of finding the smallest length for which there exists a linear code of dimension k and minimum distance d , over a field of q elements. We consider here the problem for quaternary codes( q =4), solving the problem for k ⩽ for all values of d , and for k =4 for all but ten values of d .

An optimal ternary [69, 5, 45] code and related codes

A ternary [69, 5, 45] code is constructed, thus solving the problem of finding the minimum length of a ternary code of dimension 5 and minimum distance 45. Furthermore, this code is shown to be a unique two-weight code with weight enumerator 1+210Z45+32Z54. It is also shown that a ternary [70, 6, 45] code, which would have been a projective two-weight code giving rise to a new strongly regular graph, does not exist. In order to prove the main results, the uniqueness of some other optimal ternary codes with specified weight enumerators is also established.

On Group Partitions Associated with Lower Bounds for Symmetric Ramsey Numbers

Most of the best available lower bounds for symmetric Ramsey numbers arise from partitions of abelian groups into classes which have a certain difference-free property and which, in addition, turn out to be images of each other under group automorphisms. We make a detailed study of group partitions having this latter property, and report the results of exhaustive searches for partitions of this type which yield improved lower bounds for certain of these Ramsey numbers.

Searching with lies: the Ulam problem

Abstract Ulams problem is to determine the minimal number of yes-no queries sufficient to find an unknown integer between 1 and 2 20 if at most some given number e of the answers may be lies. The problem has recently been solved for the cases e = 1 and e = 2. In this paper, we solve the problem for the cases e = 3 and e = 4. We also discuss Ulams problem in the situation where all the queries must be stated in advance and may not make use of intermediate feedback. This last problem is equivalent to one of finding optimal error-correcting codes.

The solution of a problem of Ulam on searching with lies

We consider Ulams problem of determining the minimum number of yes-no queries to find an unknown integer between 1 and 2/sup 20/ if at most some given number e of the answers may be lies. Previously published papers have solved the problem for cases e=1,2,3 and 4. In this paper we solve the problem for all values of e.