Vassili C. Mavron

Information sets and partial permutation decoding for codes from finite geometries

We determine information sets for the generalized Reed-Muller codes and use these to apply partial permutation decoding to codes from finite geometries over prime fields. We also obtain new bases of minimum-weight vectors for the codes of the designs of points and hyperplanes over prime fields.

Partial permutation decoding for codes from finite planes

We determine to what extent permutation decoding can be used for the codes from desarguesian projective and affine planes. We define the notion of s-PD-sets to correct s errors, and construct some specific small sets for s = 2 and 3 for desarguesian planes of prime order.

On symmetric nets and generalized Hadamard matrices from affine designs

Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q3, q2, q2−1/q−1) designs without symmetric (q, q)-subnets.

On the Structure of Affine Designs

An important characteristic of an affine space is that each of its hyperplanes is itself an affine space of one dimension lower. This property plays an essential part in the analysis of the structure of affine spaces. The purpose of this paper is to develop an analogous structure theory for general affine designs. This enables the introduction of a concept of dimension for affine designs and also yields techniques for constructing affine designs from smaller ones. The initial sections introduce the required basic results and terminology. In w is proved the main decomposition theorem which demonstrates how an affine design may be constructed with a given decomposition into sets of smaller affine designs. The results of this paper form part of my doctoral thesis at the University of London. To my supervisor Professor D.R. Hughes, I am indebted for invaluable assistance and guidance.

Translations and parallel classes of lines in affine designs

Abstract The structure of affine designs admitting all possible translations in one direction is investigated. A method is given for constructing such designs using Cartesian groups, analogous to the known method for affine planes.

An upper bound for the minimum weight of the dual codes of desarguesian planes

We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combin. 23 (2002) 529-538] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order p^m where p is a prime, and m>=1. This gives words of weight 2p^m+1-p^m-1p-1 in the dual of the p-ary code of the desarguesian plane of order p^m, and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of Andre planes. We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmaros and Mazzocca [Gabor Korchmaros, Francesco Mazzocca, On (q+t)-arcs of type (0,2,t) in a desarguesian plane of order q, Math. Proc. Cambridge Philos. Soc. 108 (1990) 445-459].

On affine designs and Hadamard designs with line spreads

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291-303] described a construction that relates any Hadamard design H on 4^m-1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m,4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m,4) if, and only if, H is the classical design of points and hyperplanes in PG(2m-1,2) and the line spread is of a special type. Computational results about line spreads in PG(5,2) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG(3,4), and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153-226].

Translations and construction of generalised nets

Abstract A theory of certain types of translations for generalised nets is developed, and the structure of nets constructed by a special method is analysed, especially with regard to subsets and extensions. This enables nonisomorphism results to be established, along with theorems that the number of nonisomorphic solutions of a certain type (for example, complete or maximal) tends to infinity with k .

Embeddable transversal designs

The embeddability of certain (group) divisible designs in symmetric 2-designs is investigated. These designs are symmetric resolvable transversal designs. It is proved that all such transversal designs with v = 2k are embeddable and some necessary and sufficient conditions for other cases are given.

Critical sets in nets and latin squares

Abstract The concept of a critical set in a latin square is extended to the more general setting of nets. A lower bound is given for the size of a critical set in a group-based net. In the case of a general net of degree 3 and order n ( n ⩾5), it is shown that the size of a critical set is bounded below by n +1. In the proof of this result, a special embedding of a latin square of order m into a suitable latin square of order n is established for every n >2 m .