Simon R. Blackburn

pseudo-random array is a two-dimensional array in which each n₁×n₂ nonzero matrix is contained exactly once as a window in the array. The pseudo-random arrays we consider have many desirable properties in addition, such as the shift-and-add-property, i.e., the addition of the array to any of its nontrivial shifts is another nontrivial shift of the array. A pseudorandom array code is a linear code of r₁ × r₂ arrays in which each n₁ × n2 nonzero matrix is contained exactly once as a window in one of the arrays. In this paper, new parameters for pseudo-random arrays are presented, and their construction is generalized to pseudo-random array codes. Our constructions of pseudo-random array codes are based on the folding of sequences. Two techniques to verify whether an array or a set of arrays constructed by folding is a pseudo-random array or a pseudo-random array code, respectively, are presented. These verification techniques can also be used for VLSI testing.

Let F n be a free group of rank n, with free generating set X. A subset D of F n is a Distinct Difference Configuration if the differences g-1 h are distinct, where g and h range over all (ordered) pairs of distinct elements of D. The subset D has diameter at most d if these differences all have word length at most d. When n is fixed and d is large, the paper shows that the largest distinct difference configuration in F n of diameter at most d has size approximately (2n-1) d/3 .

The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and Kötter in 2010 to model data transmission using random linear network coding. The input and output of the channel are n × m matrices over a finite field F_q. When the matrix X is input, the channel outputs Y = A(X + W) where A is a uniformly chosen n × n invertible matrix over F_q and where W is a uniformly chosen n × m matrix over F_q of rank t. Silva et al. considered the case when 2n ≤ m. They determined the asymptotic capacity of the AMMC when t, n and m are fixed and q ⟶ ∞. They also determined the leading term of the capacity when q is fixed, and t, n and m grow linearly. We generalise these results, showing that the condition 2n ≥ m can be removed. (Our formula for the capacity falls into two cases, one of which generalises the 2n ≥ m case.) We also improve the error term in the case when q is fixed.

We define and investigate the property of being “exponent-critical” for a finite group. A finite group is said to be exponent-critical if its exponent is not the least common multiple of the exponents of its proper non-abelian subgroups. We explore properties of exponent-critical groups and give a characterization of such groups. This characterization generalizes a classical result of Miller and Moreno on minimal non-abelian groups; interesting families of 𝑝-groups appear.

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.