Nabil Shalaby

Existence of Perfect 4-Deletion-Correcting Codes with Length Six

By a T*(2, k, v)-code we mean a perfect4-deletion-correcting code of length 6 over an alphabet of size v, which is capable of correcting anycombination of up to 4 deletions and/or insertions of letters that occur in transmission of codewords. Thethird author (DCC Vol. 23, No. 1) presented a combinatorial construction for such codes and prove thata T*(2, 6, v)-code exists for all positive integers v≢ 3 (mod 5), with 12 possible exceptions of v. In this paper, the notion of a directedgroup divisible quasidesign is introduced and used to show that a T*(2, 6,v)-code exists for all positive integers v ≡ 3 (mod 5), except possiblyfor v ∈ {173, 178, 203, 208}. The 12 missing cases for T*(2,6, v)-codes with v ≢ 3 (mod 5) are also provided, thereby the existenceproblem for T*(2, 6, v)-codes is almost complete.

Block disjoint difference families for Steiner triple systems: v≡3 mod 6

Abstract A block disjoint (v,k,λ) difference family is a difference family with disjoint blocks. We show that disjoint (v,3,1) difference families exist for all v≡3 mod 6 with v⩾3.

The existence of (p,q)-extended Rosa sequences

A (p,q)-extended Rosa sequence is a sequence of length 2n+2 containing each of the symbols 0,1,...,n exactly twice, and such that two occurrences of the integer j>0 are separated by exactly j-1 symbols. We prove that, with two exceptions, the conditions necessary for the existence of a (p,q)-extended Rosa sequence with prescribed positions of the symbols 0 are sufficient. We also extend the result to @l-fold (p,q)-extended Rosa sequences; i.e., the sequences where every pair of numbers is repeated exactly @l times.

Kirkman frames having hole type h u m 1 for small h

We begin an investigation into the spectrum of (non-uniform) Kirkman frames of type hum1, paying particular attention to the cases where h ∈{2,4,6,8,10,12}. We show that for each of these values of h the obvious necessary conditions on u, m are sufficient in all but at most 24 cases of (h,u,m).

Directed packing and covering designs with block size four

In this paper we improve and generalize the results of two earlier papers of Skillicorn. The maximum (minimum) number of blocks in a directed (v; 4 ;� ) packing (covering) designs are completely determined. c 2001 Elsevier Science B.V. All rights reserved.

Hamilton cycles in restricted block-intersection graphs

As part of our main result we prove that the blocks of any sufficiently large BIBD(v, 4, λ) can be circularly ordered so that consecutive blocks intersect in exactly one point, i.e., that the 1-block-intersection graphs of such designs are Hamiltonian. In fact, we prove that such graphs are Hamilton-connected. We also consider {1, 2}-block-intersection graphs, in which adjacent vertices have either one or two points in common between their corresponding blocks. These graphs are Hamilton-connected for all sufficiently large BIBD(v, k, λ) with

Skolem and Rosa rectangles and related designs

{k \in \{4,5,6\}}

The existence of near-Rosa and hooked near-Rosa sequences

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