T. P. McDonough

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.

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].

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 .

Symmetric designs and geometroids

Aλ-setS in a symmetric 2-(v, k, λ) designΠ is a subset which every block meets in 0, 1 orλ points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples ofλ-secs. It is shown that ifπ has aλ-set thenπ is a geometroid withv=λu2+u+1 andk=λu+1, whereu≧λ−1. The cases whenu isλ−1,λ andλ+1 are investigated and some open problems discussed.

Reed-Muller codes and permutation decoding

We show that the first- and second-order Reed-Muller codes, R(1,m) and R(2,m), can be used for permutation decoding by finding, within the translation group, (m-1)- and (m+1)-PD-sets for R(1,m) for m>=5,6, respectively, and (m-3)-PD-sets for R(2,m) for m>=8. We extend the results of Seneviratne [P. Seneviratne, Partial permutation decoding for the first-order Reed-Muller codes, Discrete Math., 309 (2009), 1967-1970].

Quasi-Symmetric Designs with Good Blocks and Intersection Number One

We show that a quasi-symmetric design with intersection numbers 1 and y > 1 and a good block belongs to one of three types: (a) it has the same parameters as PG2(4, q), the design of points and planes in projective 4-space; (b) it is the 2-(23, 7, 21) Witt design; (c) its parameters may be written v = 1 + ((α − 1)λ + 1)(y − 1) and k = 1 + α(y − 1), where α is an integer and α > y ≥ 5, and the design induced on a good block is a 2-(k, y, 1) design. No design of type (c) is known; moreover, for large ranges of the parameters, it cannot exist for arithmetic reasons concerning the parameters. We show also that PG2(4, q) is the only design of type (a) in which all blocks are good.

The Geometry of Frequency Squares

This paper establishes a correspondence between mutually orthogonal frequency squares (MOFS) and nets satisfying an extra property (“framed nets”). In particular, we provide a new proof for the bound on the maximal size of a set of MOFS and obtain a geometric characterisation of the case of equality: necessary and sufficient conditions for the existence of a complete set of MOFS are given in terms of the existence of a certain type of PBIBD based on the L2-association scheme. We also discuss examples obtained from classical affine geometry and recursive construction methods for (complete) sets of MOFS.

Improved partial permutation decoding for Reed-Muller codes

It is shown that for n ź 5 and r ź n - 1 2 , if an ( n , M , 2 r + 1 ) binary code exists, then the r th-order Reed-Muller code R ( r , n ) has s -PD-sets of the minimum size s + 1 for 1 ź s ź M - 1 , and these PD-sets correspond to sets of translations of the vector space F 2 n . In addition, for the first order Reed-Muller code R ( 1 , n ) , s -PD-sets of size s + 1 are constructed for s up to the bound ź 2 n n + 1 ź - 1 . The results apply also to generalized Reed-Muller codes.