arXiv subjects
Publications and source records attributed to Austin Allen.
In this article we improve the dimension and minimum distance bound of the the Hermitian Lifted Codes LRCs construction from L\'opez, Malmskog, Matthews, Pi\~nero and Wooters (L\'opez et. al.) via elementary univariarte polynomial division. They gave an asymptotic rate estimate of $0.007$. For the case where $q$ is a power of $2$ we improve the rate estimate to $0.010$ using univariate polynomial division.
Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.
In this article we give an order-dividing bijective function between cyclic and non cyclic groups of finite order. In particular, we prove that there exists a bijective function from D_{2n} to Z_{2n} for any natural integer n; and from Z_p x Z_k to Z_{pk} when p is an odd prime and k is not a multiple of p.