A class of Steiner 4-wise balanced designs derived from preparata codes
✍ Scribed by Vladimir D. Tonchev
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 119 KB
- Volume
- 4
- Category
- Article
- ISSN
- 1063-8539
No coin nor oath required. For personal study only.
✦ Synopsis
The minimum weight codewords in the Preparata code of length n = 4" are utilized for the construction of an infinite family of Steiner S(4, {5,6}, 4"l + 1) designs for any rn 2 2. 0 1996 John Wiley & Sons, Inc.
A t-wise balanced design with parameters t -(w, Ic, A) is a pair (X, 0) where X is a set of points and , L? is a collection of subsets of X (called blocks) with sizes from the set Ic, such that every t-subset of X is contained in exactly X blocks. If 1x1 = 1, that is, all blocks are of the same size, say k , the design is a t -(TI, k , A) design. A Steiner design (or system) is a design with X = 1. The notation S ( t , Ic, w)
[resp. S ( t , k , A)] is often used in this case. There has been recent interest in Steiner twise balanced designs, motivated by the lack of any known infinite family of Steiner systems S ( t , k , w) for t 2 4 [31.
Kramer and Mathon
[3] studied t-wise balanced Steiner designs on w 5 16 points and their extensions. One interesting design found in [3] is an S(4, {5,6}, 17), considered by the authors as a closest approximation of a yet unknown (and probably nonexistent) Steiner system S(4,5,17). Note that the S(4, {5,6}, 17) design is unique (up to isomorphism) for the given parameters [3].
The aim of this note is to describe an infinite class of Steiner 4-wise balanced designs containing the S(4, { 5 , 6 } , 17) from [3] as the smallest example. Namely, the minimum weight codewords in the Preparata code of length n = 4m are used for the construction of an S(4, ( 5 , 6}, 4" + 1) for any m 2 2. The smallest design S(4, { 5 , 6 } , 17) ( m = 2 ) corresponds to the Nordstrom-Robinson code. The construc-