Further results on error correcting binary group codes

The present paper is a sequel to the paper “On a class of error-correcting binary group codes”, by R. C. Base and D. K. Ray-Chaudhuri, appearing in Information and Control in which an explicit method of constructing a t-error correcting binary group code with n = 2m − 1 places and k = 2m − 1 − R(m,t) ≧ 2m − 1 − mt information places is given. The present paper generalizes the methods of the earlier paper and gives a method of constructing a t-error correcting code with n places for any arbitrary n and k = n − R(m,t) ≧ [(2m − 1)/c] − mt information places where m is the least integer such that cn = 2m − 1 for some integer c. A second method of constructing t-error correcting codes for n places when n is not of the form 2m − 1 is also given

Embedding of a pseudo-residual design into a Möbius plane

AbstractLet U be a class of subsets of a finite set X. Elements of U are called blocks. Let v, t and λ1, 0 ⩽ i ⩽ t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ0, λ1,…, λt; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λt blocks, 0 ⩽ i ⩽ t, and (3) for every block A in U, |A| ϵ K. It is well-known that if K consists of a singleton k, then λ0,…, λt − 1 can be determined from υ, t, k and λt. Hence, we shall denote a (λ0,…, λt; {k}, υ)t-design by Sλ(t, k, υ), where λ = λt. A Möbius plane M is an S1(3, q + 1, q2 + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ0 = q3 + q − 1, λ1 = q2 + q, λ2 = q + 1, λ3 = 1, K = {q + 1, q, q − 1}, and υ = q2 − 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition

On resolvable designs

AbstractA balanced incomplete block design (BIBD) B[k, λ;v] is an arrangement of v elements in blocks of k elements each, such that every pair of elements is contained in exactly λ blocks. A BIBD B[k, 1;v] is called resolvable if the blocks can be partitioned into (v−1)(k−1) families each consisting of v/k mutually disjoint blocks. Ray-Chaudhuri and Wilson [8] proved the existence of resolvable BIBD's B[3, 1; v] for every v≡3 (mod 6). In addition to this result, the existence is proved here of resolvable BIBD's B[4, 1; v] for every v≡4 (mod 12)

Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc

On a class of error correcting binary group codes

A general method of constructing error correcting binary group codes is obtained. A binary group code with n places, k of which are information places is called an (n,k) code. An explicit method of constructing t-error correcting (n,k) codes is given for n = 2m−1 and k = 2m−1−R(m,t) ≧ 2m−1−mt where R(m,t) is a function of m and t which cannot exceed mt. An example is worked out to illustrate the method of construction