We initiate the study of λ-fold near-factorizations of groups with λ>1. While λ-fold near-factorizations of groups with λ=1 have been studied in numerous papers, this is the first detailed treatment for λ>1. We establish fundamental properties of λ-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of λ-fold near-factorizations, including upper bounds on λ. We present three constructions of infinite families of λ-fold near-factorizations, highlighting the characterization of two subfamilies of λ-fold near-factorizations. We discuss a computational approach to λ-fold near-factorizations and tabulate computational results for abelian groups of small order.

Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which over-laps of certain specified sizes are forbidden. We prove some general bounds and we give several constructions in the case of binary codes. Our techniques also allow us to provide an alternative, elementary proof of a lower bound on non-overlapping codes due to Levenshtein [9] in 1964.

For each prime power $n \equiv 1$ (mod 4), a pair of connected graphs on 4n-4 vertices, with reconstruction number at least 2n-1, is constructed.