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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. Index Terms--Constant-composition codes, group divisible codes, group divisible designs, recursive constructions.

This correspondence introduces two new constructive techniques to complete the determination of the sizes of optimal q-ary codes of constant weight three and distance four. Index Terms--Constant-weight codes, large sets with holes, sequences.

The radar array problem arises from the need to design frequency hopping sequences with small out-of-phase autocorrelations. It assumes the reflected signals have negligible Doppler shifts, so the correlations are calculated along the time axis only. In this correspondence, a systematic construction for radar arrays is provided by means of homogeneous uniform difference matrices. A systematic construction for properly centered permutation matrices, a special kind of homogeneous uniform difference matrices, is also provided, which partially solves the open problems posed by Zhang and Tu. Index Terms--Homogeneous uniform difference matrix, partitioned directed difference packing, properly centered permutation matrices, radar array.

We show an interesting pairwise balanced design (PBD)-closure result for the set of lengths of constant-composition codes whose distance and size meet certain conditions. A consequence of this PBD-closure result is that the size of optimal constant-composition codes can be determined for infinite families of parameter sets from just a single example of an optimal code. As an application, the size of several infinite families of optimal constant-composition codes are derived. In particular, the problem of determining the size of optimal constant-composition codes having distance four and weight three is solved for all lengths sufficiently large. This problem was previously unresolved for odd lengths, except for lengths seven and eleven. Index Terms--Constant-composition codes, group divisible designs, pairwise balanced designs (PBD), PBD-closure.

We develop a connection between permutation arrays that are used in powerline communication and well-studied combinatorial objects, mutually orthogonal latin squares (MOLS). From this connection, many new results on permutation arrays can be obtained. Index Terms--Doubly resolvable design, mutually orthogonal latin squares (MOLS), permutation array, permutation code, powerline communications.

This correspondence obtains new difference triangle sets using affine planes. Index Terms--Affine difference sets, difference triangle sets.