2,875 research outputs found
Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three
The concept of group divisible codes, a generalization of group divisible
designs with constant block size, is introduced in this paper. This new class
of codes is shown to be useful in recursive constructions for constant-weight
and constant-composition codes. Large classes of group divisible codes are
constructed which enabled the determination of the sizes of optimal
constant-composition codes of weight three (and specified distance), leaving
only four cases undetermined. Previously, the sizes of constant-composition
codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table
Hadamard matrices modulo p and small modular Hadamard matrices
We use modular symmetric designs to study the existence of Hadamard matrices
modulo certain primes. We solve the -modular and -modular versions of
the Hadamard conjecture for all but a finite number of cases. In doing so, we
state a conjecture for a sufficient condition for the existence of a
-modular Hadamard matrix for all but finitely many cases. When is a
primitive root of a prime , we conditionally solve this conjecture and
therefore the -modular version of the Hadamard conjecture for all but
finitely many cases when , and prove a weaker result for
. Finally, we look at constraints on the existence of
-modular Hadamard matrices when the size of the matrix is small compared to
.Comment: 14 pages; to appear in the Journal of Combinatorial Designs; proofs
of Lemma 4.7 and Theorem 5.2 altered in response to referees' comment
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