Jennifer D. Key

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.

Permutation decoding for the binary codes from triangular graphs

By finding explicit PD-sets we show that permutation decoding can be used for the binary code obtained from an adjacency matrix of the triangular graph T(n) for any n ≥ 5.

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.

Minimum-weight codewords as generators of generalized Reed-Muller codes

We establish the range of values of /spl rho/, where 0/spl les//spl rho//spl les/m(q-1), for which the generalized Reed-Muller code R/sub Fq/(/spl rho/, m) of length q/sup m/ over the field F/sub q/ of order q is spanned by its minimum-weight vectors.

Hadamard matrices and their designs: a coding-theoretic approach

Given an m x m Hadamard matrix one can extract m2 symmetric designs on m - 1 points each of which extends uniquely to a 3-design. Further, when m is a square, certain Hadamard matrices yield symmetric designs on m points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary associa- tion of linear codes with designs. This leads naturally to the notion, defined for any prime p , of p-equivalence for Hadamard matrices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty 24 x 24 matrices fall into only six 2-equivalence classes. In the 16x16 case, 2-equivalence is identical to the standard equivalence, but our results illu- minate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian 2-group, why two of the matrices cannot be obtained from a symmetric design on 16 points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.

Minimum Weight and Dimension Formulas for Some GeometricCodes

The geometric codes are the duals of the codes defined by the designs associated with finite geometries. The latter are generalized Reed–Muller codes, but the geometric codes are, in general, not. We obtain values for the minimum weight of these codes in the binary case, using geometric constructions in the associated geometries, and the BCH bound from coding theory. Using Hamadas formula, we also show that the dimension of the dual of the code of a projective geometry design is a polynomial function in the dimension of the geometry.

Codes from incidence matrices and line graphs of Hamming graphs

We examine the p-ary codes, for any prime p, that can be obtained from incidence matrices and line graphs of the Hamming graphs, H(n,m), obtaining the main parameters of these codes. We show that the codes from the incidence matrices of H(n,m) can be used for full permutation decoding for all m,n>=3.

Affine and projective planes

The aim of this work is to suggest a setting for the discussion and classification of finite projective planes. Our approach rests heavily on the results and techniques of algebraic coding theroy. The approach is to study a finite projective plane II via its various affine parts and, to this end, we introduce the notion of the hull of an affine plane π: the hull turns out to be the code generated, over an appropriate finite field F p , by all differences of those pairs of rows of an indicence matrix that represent parallel lines of π

Steiner triple systems with doubly transitive automorphism groups: A corollary to the classification theorem for finite simple groups

Assuming that the classification theorem for finite simple groups is complete, a conjecture of M. Hall (Proc. Sympos. Pure Math. 6 (1962), 47–66; and in “Proceedings of the International Conference on Theory of Groups”, pp. 115–144, Australian National University, Canberra, Australia, 1965) that a Steiner triple system with a doubly transitive automorphism group is a projective or affine geometry, is verified.

Codes associated with triangular graphs and permutation decoding

Linear codes that can be obtained from designs associated with the complete graph on n vertices and its line graph, the triangular graph, are examined. The codes have length n choose 2, dimension n or n − 1, and minimum weight n − 1 or 2n − 4. The parameters of the codes and their automorphism groups for any odd prime are obtained and PD-sets inside the symmetric group S n are found for full permutation decoding for all primes and all integers n ≥ 6.