Ternary Golay Code Research Articles

A sublattice of the Leech lattice

In this paper a rank 12 even lattice C is constructed which is type 3 (if v ϵ C then ( v , v ) ⩾ 6), and which has the following maximality property: C ⊕ C + 3 · Λ 24 can be embedded in the Leech lattice Λ 24 , and is a maximal type 3 sublattice of Λ 24 . The construction uses properties of the binary and ternary Golay codes.

The uniqueness of the near hexagon on 729 points

We prove that any regular near hexagon with 729 vertices and lines of size 3 is derived from the ternary Golay code, thus settling the last case in doubt among the regular near hexagons with lines of size 3.

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.

On the partial geometry pg(6, 6, 2)

We give some new representations of the partial geometry pg(6, 6, 2), which was constructed by van Lint and Schrijver, show a connection with a strongly regular graph based on the ternary Golay code, and determine the automorphism group of the geometry.

Split codes and the Mathieu groups

The aim of this article is to present a variation on the theme of the Mathieu groups as automorphism groups of codes. The original theme is exposed in the survey article of Conway [2]. There it is shown that the Mathieu groups MS4 and Ml, form the main constituents of the automorphism groups of the binary and ternary Golay codes. This paper is a study of the mechanism which forces the Golay codes to have such large automorphism groups. It is well known that the Golay codes are examples of the class of codes called quadratic residue codes. We introduce the notion of split quadratic residue codes and show that the Golay codes can also be viewed as split codes. We can then conclude that the permutation group of each Golay code must contain two distinct groups of PSL type and must therefore be quite large. We obtain the isomorphisms between the ordinary and the split codes explicitly. Thus, we can show exactly how the two PSL groups act in the permutation group of each Golay code. The split codes defined here overlap with certain codes studied by Karlin [6] and Pless [9]. In fact, the two isomorphisms which we mentioned above are discussed by Karlin and Pless: Karlin treats the binary Golay code and Pless treats the ternary Golay code. In our approach, both isomorphisms are studied in a uniform fashion. We are also able to find one additional isomorphism of this type. The Parker-Nikolai study [7] makes it seem likely that we have found all of the isomorphisms of this particular kind. The paper is divided into 8 sections titled as follows:

Unrestricted codes with the golay parameters are unique

The paper contains a proof of the uniqueness of both binary and ternary Golay codes, without assumption of linearity. Similar results are obtained about the extended and expurgated Golay codes. The method consists in proving the linearity, which, according to Pless' results, implies the uniqueness.

The regular two-graph on 276 vertices

There is a unique regular two-graph on 276 vertices. This provides a characterization of Conway's group ⋯ 3. The proof is based on 276 ≃ 3 × 11 + 3 5, and uses the ternary Golay code. The paper contains a list of the known strongly regular graphs with the eigenvalue −5.