Generalized Steiner Systems Research Articles

Constructions of optimal quaternary constant weight codes via group divisible designs

Generalized Steiner systems GS ( 2 , k , v , g ) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g + 1 with minimum Hamming distance 2 k − 3 , in which each codeword has length v and weight k . As to the existence of a GS ( 2 , k , v , g ) , a lot of work has been done for k = 3 , while not so much is known for k = 4 . The notion k - ∗ GDD was first introduced by Chen et al. and used to construct GS ( 2 , 3 , v , 6 ) . The necessary condition for the existence of a 4 - ∗ GDD ( 6 v ) is v ≥ 14 . In this paper, it is proved that there exists a 4 - ∗ GDD ( 6 v ) for any prime power v ≡ 3 , 5 , 7 ( mod 8 ) and v ≥ 19 . By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.

Constructions for generalized Steiner systems GS(3, 4, v, 2)

Generalized Steiner systems GS (3, 4, v, 2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance three, in which each codeword has length v and weight four. Not much is known for GS (3, 4, v, 2)s except for a recursive construction and two small designs for v = 8,10 given by Etzion. In this paper, more small designs are found by computer search and also given are direct constructions based on finite fields and rotational Steiner quadruple systems and recursive constructions using three-wise balanced designs. Some infinite families are also obtained.

Existence of Generalized Steiner Systems GS(2,4,ν,2)

Generalized Steiner systems GS(2,4,v,2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance five, in which each codeword has length v and weight four. Etzion conjectured its existence for any integer v? 7 and v? 1(mod 3). The conjecture has been verified for prime powers v > 7 and v? 7(mod 12) by the latter two of the present authors. It has also been pointed out that there does not exist a GS(2,4,7,2). In this paper, constructions using frame generalized Steiner systems, two holey perfect bases and orthogonal Latin squares are discussed. With these constructions the conjecture is confirmed with the exception for v=7 and three possible exceptions for v ?13, 52, 58.

Some new optimal quaternary constant weight codes

Constant weight codes (CWCs) are an important class of codes in coding theory. Generalized Steiner systems GS(2, k , v , g ) were first introduced by Etzion and used to construct optimal nonlinear CWCs over an alphabet of size g +1 with minimum Hamming distance 2 k −3, in which each codeword has length v and weight k . In this paper, Weil's theorem on character sum estimates is used to show that there exists a GS(2,4, v ,3) for any prime v ≡ 1 ( mod 4) and v >13. From the coding theory point of view, an optimal nonlinear quaternary ( v ,5,4) CWC exists for such a prime v .

Generalized Steiner systems GS4 (2, 4, v, g) for g = 2, 3, 6

AbstractGeneralized Steiner systems GSd(t, k, v, g) were first introduced by Etzion and used to construct optimal constant‐weight codes over an alphabet of size g + 1 with minimum Hamming distance d, in which each codeword has length v and weight k. Much work has been done for the existence of generalized Steiner triple systems GS(2, 3, v, g). However, for block size four there is not much known on GSd(2, 4, v, g). In this paper, the necessary conditions for the existence of a GSd(t, k, v, g) are given, which answers an open problem of Etzion. Some singular indirect product constructions for GSd(2, k, v, g) are also presented. By using both recursive and direct constructions, it is proved that the necessary conditions for the existence of a GS4(2, 4, v, g) are also sufficient for g = 2, 3, 6. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 401–423, 2001

Generalized Steiner Systems {\rm GS}(2, 4, v, 2) with v a Prime Power \equiv 7 ({\it mod}\ {12})

Generalized Steiner systems {\rm GS}(2, 4, v, g) were first introduced by Etzion and were used to construct optimal constant weight codes over an alphabet of size g+1 with minimum Hamming distance 5, in which each codeword has length v and weight 4. Etzion conjectured that the necessary conditions v\equiv 1 ({\it mod}\ {3}) and v \geq 7 are also sufficient for the existence of a {\rm GS}(2,4,v,2). Except for the example of a {\rm GS}(2,4,10,2) and some recursive constructions given by Etzion, nothing else is known about this conjecture. In this paper, Weil's theorem on character sum estimates is used to show that the conjecture is true for any prime power v\equiv 7 ({\it mod}\ {12}) except v=7, for which there does not exist a {\rm GS}(2,4,7,2).

Generalized steiner triple systems with group size five

Generalized Steiner triple systems, GS(2, 3, n, g) are used to construct maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The existence of GS(2, 3, n, g) has been solved for g=2, 3, 4, 9. In this paper, by introducing a special kind of holey generalized Steiner triple systems (denoted by HGS(2, 3, (n, u), g)), singular indirect product (SIP) construction for GDDs is used to construct generalized Steiner systems. The numerical necessary conditions for the existence of a GS(2, 3, n, g) are shown to be sufficient for g=5.

Constant Weight Codes and Group Divisible Designs

The study of a class of optimal constant weight codes over arbitrary alphabets was initiated by Etzion, who showed that such codes are equivalent to special GDDs known as generalized Steiner systems GS(t,k,n,g) Etzion. This paper presents new constructions for these systems in the case t=2, k=3. In particular, these constructions imply that the obvious necessary conditions on the length n of the code for the existence of an optimal weight 3, distance 3 code over an alphabet of arbitrary size are asymptotically sufficient.

On the Number of Blocks in a Generalized Steiner System

We considert-designs withλ=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. Fort=2, this inequality is the well known De Bruijn–Erdős inequality. Fort>2 it has the same order of magnitude as the Wilson–Petrenjuk inequality for Steiner systems with constant block size. The point of this note is that the inequality is very easy to derive and does not seem to be known. A stronger inequality was derived in 1969 by Woodall (J. London Math. Soc.(2)1, 509–519), but it requires Lagrange multipliers in the proof.

Optimal constant weight codes over Zk and generalized designs

We consider optimal constant weight codes over arbitrary alphabets. Some of these codes are derived from good codes over the same alphabet, and some of these codes are derived from block design. Generalizations of Steiner systems play an important role in this context. We give several construction methods for these generalizations. An interesting class of codes are those which form generalized Steiner systems and their supports form ordinary Steiner systems. Finally, we consider classes of codes which are MDS constant weight codes.

Generalized Steiner systems with block size three and group sizeg ? 3(mod 6)

Generalized Steiner systems with block size three and group sizeg ? 3(mod 6)

The (16,6,2) Biplane with 60 Ovals and the Weight Distribution of a Code

Structural relations of the unique (16, 6, 2) biplane with 60 ovals are given. These and a theorem of Assmus and Novillo on homogeneous generalized Steiner systems of type 3-{ v , {4, 6}, 1) allow for a proof of the fact that the code equal to the row span, over GF(2), of the incidence matrix of a 3-(112, 12, 1) design does not contain weight-16 vectors.

On Generalized Steiner Systems and Semi-biplanes

An exhaustive computer search has established the existence of precisely two new homogeneous 3-(20,{4,6},1) generalized Steiner systems whose 6-subsets form a semi-biplane. These designs are described by means of their automorphism groups.