Upper bounds on the minimum Lee distance of codes that are linear over Zq,q=pt,pprime are discussed. The bounds are Singleton like, depending on the length, rank, and alphabet size of the code. Codes meeting such bounds are referred to as Maximum Lee Distance with respect to Rank (MLDR) Codes. We present some new bounds on MLDR codes, using combinatorial arguments. In the context of MLDR codes, our work provides improvements over existing bounds in the literature.

The weight spectra of MDS codes of length n and dimension k over arbitrary alphabets are studied. For all q-ary MDS codes of dimension k and length n ≠ q + k - 1 containing the zero codeword, it is shown that all k weights from n to n - k + 1 are realized. The weight spectrum in the remaining case n = q + k - 1 is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained.

Several new constructions of 3-D optical orthogonal codes are presented here. In each case, the codes have ideal OFF-peak autocorrelation λa= 0, and in all, but one case a cross correlation of λc= 1. All codes produced are optimal with respect to the applicable Johnson bound either presented or developed here. Thus, on one hand the codes are as large as possible, and on the other, the bound(s) are shown to be tight. All codes are constructed by using a particular automorphism (a singer cycle) of PG(k, q), the finite projective geometry of dimension k over the field of order q, or by using an affine analogue in AG(k, q).

We provide a new construction yielding one new and one known infinite family of optimal (n,w,2) -optical orthogonal codes,wisin {4,6}. Our construction relies on various techniques in finite projective spaces involving hyperovals in projective planes and Singer groups.

The link between finite geometry and various classes of error-correcting codes is well known. Arcs in projective spaces, for instance, have a close tie to linear MDS codes as well as the high-performing low-density parity-check codes. In this article, we demonstrate a connection between arcs and optical orthogonal codes (OOCs), a class of nonlinear binary codes used for many modern communication applications. Using arcs and Baer subspaces of finite projective spaces, we construct some infinite classes of OOCs with auto-correlation and cross-correlation both larger than 1.