There exist Steiner triple systems of order 15 that do not occur in a perfect binary one‐error‐correcting code

The codewords at distance three from a particular codeword of a perfect binary one‐error‐correcting code (of length 2^m?1) form a Steiner triple system. It is a longstanding open problem whether every Steiner triple system of order 2^m?1 occurs in a perfect code. It turns out that this is not the case; relying on a classification of the Steiner quadruple systems of order 16 it is shown that the unique anti‐Pasch Steiner triple system of order 15 provides a counterexample. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 465–468, 2007     

Weighted Hamming metric structures

It is known that the binary generalized Goppa codes are perfect codes for the weighted Hamming metrics. In this paper, we present the existence of a weighted Hamming metric that admits a binary Hamming code (resp. an extended binary Hamming code) to be perfect code. For a special weighted Hamming metric, we also give some structures of a 2-perfect code, show how to construct a 2-perfect linear code and obtain the weight distribution of a 2-perfect code from the partial information of the code.     

Configuration distribution and designs of codes in the Johnson scheme

The main goal of this article is to present several connections between perfect codes in the Johnson scheme and designs, and provide new tools for proving Delsarte conjecture that there are no nontrivial perfect Codes in the Johnson scheme. Three topics will be considered. The first is the configuration distribution which is akin to the weight distribution in the Hamming scheme. We prove that if there exists an e‐perfect code in the Johnson scheme then there is a formula which connects the number of vectors at distance i from any codeword in various codes isomorphic to . The second topic is the Steiner systems embedded in a perfect code. We prove a lower bound on the number of Steiner systems embedded in a perfect code. The last topic is the strength of a perfect code. We show two new methods for computing the strength of a perfect code and demonstrate them on 1‐perfect codes. We further discuss how to settle Delsarte conjecture. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 15–34, 2007     

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary (n?=?2 ^m ? 3, 2 ^n-m-1, 4) code C, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the n-cube into six cells. An arbitrary binary (n?=?2 ^m ? 4, 2 ^n-m , 3) code D, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) with six cells. As a corollary, the codes C and D are completely semiregular; i.e., the weight distribution of such codes depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if D is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable.     

A construction of mutually disjoint Steiner systems from isomorphic Golay codes

It is well known that the extended binary Golay [24,12,8] code yields 5-designs. In particular, the supports of all the weight 8 codewords in the code form a Steiner system S(5,8,24). In this paper, we give a construction of mutually disjoint Steiner systems S(5,8,24) by constructing isomorphic Golay codes. As a consequence, we show that there exists at least 22 mutually disjoint Steiner systems S(5,8,24). Finally, we prove that there exists at least 46 mutually disjoint 5-(48,12,8) designs from the extended binary quadratic residue [48,24,12] code.     

On non-antipodal binary completely regular codes

Binary non-antipodal completely regular codes are characterized. Using a result on nonexistence of nontrivial binary perfect codes, it is concluded that there are no unknown nontrivial non-antipodal completely regular binary codes with minimum distance d?3. The only such codes are halves and punctured halves of known binary perfect codes. Thus, new such codes with covering radius ρ=6 and 7 are obtained. In particular, a half of the binary Golay [23,12,7]-code is a new binary completely regular code with minimum distance d=8 and covering radius ρ=7. The punctured half of the Golay code is a new completely regular code with minimum distance d=7 and covering radius ρ=6. The new code with d=8 disproves the known conjecture of Neumaier, that the extended binary Golay [24,12,8]-code is the only binary completely regular code with d?8. Halves of binary perfect codes with Hamming parameters also provide an infinite family of binary completely regular codes with d=4 and ρ=3. Puncturing of these codes also provide an infinite family of binary completely regular codes with d=3 and ρ=2. Both these families of codes are well known, since they are uniformly packed in the narrow sense, or extended such codes. Some of these completely regular codes are new completely transitive codes.     

On enumeration of nonequivalent perfect binary codes of length 15 and rank 15

All nonequivalent perfect binary codes of length 15 and rank 15 are constructed that are obtained from the Hamming code H ¹⁵ by translating its disjoint components. Also, the main invariants of this class of codes are determined such as the ranks, dimensions of kernels, and orders of automorphism groups.     

The smallest non-derived steiner triple system is simple as a loop

The smallest non-derived triple system is simple as a loop. THEOREM.If A, B are Steiner loops, and f:A→B is a homomorphism, then if B and f ^?1 (1) are derivable from Steiner quadruple systems, then so is A.     

Constructions for rotational Steiner quadruple systems

A direct construction for rotational Steiner quadruple systems of order p+ 1 having a nontrivial multiplier automorphism is presented, where p≡13 (mod24) is a prime. We also give two improved product constructions. By these constructions, the known existence results of rotational Steiner quadruple systems are extended. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 353–368, 2009     

New uniqueness proofs for the (5, 8, 24), (5, 6, 12) and related Steiner systems

New elementary proofs of the uniqueness of certain Steiner systems using coding theory are presented. In the process some of the codes involved are shown to be unique.The uniqueness proof for the (5, 8, 24) Steiner system is due to John Conway. The blocks of the system are used to generate a length 24 binary code. Any two such codes are then shown to be equivalent up to a permutation of the coordinates. This code turns out to be the extended Golay code.In the uniqueness proof for the (4, 7, 23) system, the blocks generate a length 23 code which is extended to a length 24 code. The minimum weight vectors of this larger code hold a (5, 8, 24) Steiner system. This result together with the previous one completes the proof. At this point it is also possible to conclude that the codes involved are unique and hence equivalent to the binary perfect Golay code and its extension.Continuing with the uniqueness result for the (3, 6, 22) Steiner system, the blocks generate a length 22 code which is extended to the same length 24 code by the addition of two coordinates and one additional vector. This extension ultimately requires the computation of the coset weight distribution of the length 22 code, a result heretofore unknown. The complete coset weight distribution for a specific (22, 11, 6) self-dual code is computed using the CAMAC computer system.The (5, 6, 12) and (4, 5, 11) Steiner systems are treated differently. It is shown that each system is completely determined by the choice of six blocks which may be assumed to lie in any such design. These six blocks in fact form a basis for length 12 (and 11) ternary codes corresponding to the two systems and may be generated by an algorithm independent of the designs. This algorithm is presented and the minimum weight vectors of the resulting codes, the perfect ternary Golay code and its extension, are calculated by the CAMAC system.     

A new existence proof for Steiner quadruple systems

A Steiner quadruple system of order v is an ordered pair ${(X, \mathcal{B})}$ , where X is a set of cardinality v, and ${\mathcal{B}}$ is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. Such designs exist if and only if ${v \equiv 2,4\, (\bmod\, 6)}$ . The first and second proofs of this result were given by Hanani in 1960 and in 1963, respectively. All the existing proofs are rather cumbersome, even though simplified proofs have been given by Lenz in 1985 and by Hartman in 1994. To study Steiner quadruple systems, Hanani introduced the concept of H-designs in 1963. The purpose of this paper is to provide an alternative existence proof for Steiner quadruple systems via H-designs of type 2 ⁿ . In 1990, Mills showed that for n > 3, n ≠ 5, an H-design of type g ⁿ exists if and only if ng is even and g(n ? 1)(n ? 2) is divisible by 3, where the main context is the complicated existence proof for H-designs of type 2 ⁿ . However, Mill’s proof was based on the existence result of Steiner quadruple systems. In this paper, by using the theory of candelabra systems and H-frames, we give a new existence proof for H-designs of type 2 ⁿ independent of the existence result of Steiner quadruple systems. As a consequence, we also provide a new existence proof for Steiner quadruple systems.     

The poset structures admitting the extended binary Golay code to be a perfect code

Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.     

The Existence of Augmented Resolvable Candelabra Quadruple Systems with Three Even Groups

Augmented candelabra quadruple systems play an important role in the construction of Steiner 3-designs. In this paper, we consider augmented resolvable candelabra quadruple systems with three even groups and show that the necessary conditions on the existence of ARCQS(g ³ : s) when g is even are also sufficient.     

Perfect binary codes of infinite length

A subset C of infinite-dimensional binary cube is called a perfect binary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in C are pairwise disjoint and their union cover this binary cube. Similarly, we can define a perfect binary code in zero layer, consisting of all vectors of infinite-dimensional binary cube having finite supports. In this article we prove that the cardinality of all cosets of perfect binary codes in zero layer is the cardinality of the continuum. Moreover, the cardinality of all cosets of perfect binary codes in the whole binary cube is equal to the cardinality of the hypercontinuum.     

An improved product construction of rotational Steiner quadruple systems

An improved product construction is presented for rotational Steiner quadruple systems. Direct constructions are also provided for small orders. It is known that the existence of a rotational Steiner quadruple system of order υ+1 implies the existence of an optimal optical orthogonal code of length υ with weight four and index two. New infinite families of orders are also obtained for both rotational Steiner quadruple systems and optimal optical orthogonal codes. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 433–443, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10025     

Codes of Steiner triple and quadruple systems

The code over a finite fieldF _q of orderq of a design is the subspace spanned by the incidence vectors of the blocks. It is shown here that if the design is a Steiner triple system on points, and if the integerd is such that 2 ^d –1<2 ^d+1–1, then the binary code of the design contains a subcode that can be shortened to the binary Hamming codeH _d of length 2 ^d –1. Similarly the binary code of any Steiner quadruple system on +1 points contains a subcode that can be shortened to the Reed-Muller code (d–2,d) of orderd–2 and length 2 ^d , whered is as above.