Related papers: New Constructions of Permutation Arrays
Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating…
Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most…
Given a generator set $S$ of the symmetric group ${\rm{Sym}}_n$, every permutation $\pi\in {\rm{Sym}_n}$ is a word (product of elements) of $S$. A positive integer $d(\pi)$ is associated with each $\pi\in{\rm{Sym}_n}$ taking the length of…
Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six…
We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of…
Let $X=X(n,q)$ be the set of $n\times n$ Hermitian matrices over $\mathbb{F}_{q^2}$. It is well known that $X$ gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study $d$-codes in this…
A classical recursive construction for mutually orthogonal latin squares (MOLS) is shown to hold more generally for a class of permutation codes of length $n$ and minimum distance $n-1$. When such codes of length $p+1$ are included as…
We define a statistic on the graph of commutation classes of a permutation of the symmetric group which is used to show that these graphs are equipped with a ranked poset structure, with a minimum and maximum. This characterization also…
Permutation decoding is a technique which involves finding a subset $S$, called PD-set, of the permutation automorphism group of a code $C$ in order to assist in decoding. An explicit construction of $\left \lfloor{\frac{2^m-m-1}{1+m}}…
The Pancake graph($P_n$) represents the group of all permutations on n elements, namely $S_n$, with respect to the generating set containing all prefix reversals. The diameter of a graph is the maximum of all distances on the graph, where…
Given a real number beta>1, a permutation pi of length n is realized by the beta-shift if there is some x in [0,1] such that the relative order of the sequence x,f(x),...,f^{n-1}(x), where f(x) is the factional part of beta*x, is the same…
The Cayley distance between two permutations $\pi, \sigma \in S_n$ is the minimum number of \textit{transpositions} required to obtain the permutation $\sigma$ from $\pi$. When we only allow adjacent transpositions, the minimum number of…
In this note we present a simplified and slightly generalized version of a lemma the authors published in 1987. The lemma as stated here asserts that if the order of a permutation of $n$ elements is greater than $n^{\alpha}$ then some…
A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A square-free permutation of length $n$ is $P$-crucial, where $P$ is a subset of $\{0,1,\ldots,n\}$, if any of its…
The permutation language $P_n$ consists of all words that are permutations of a fixed alphabet of size $n$. Using divide-and-conquer, we construct a regular expression $R_n$ that specifies $P_n$. We then give explicit bounds for the length…
Permutons, which are probability measures on the unit square $[0, 1]^2$ with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a $d$-dimensional generalization of these measures for all…
A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations…
An $(m,n,R)$-de Bruijn covering array (dBCA) is a doubly periodic $M \times N$ array over an alphabet of size $q$ such that the set of all its $m \times n$ windows form a covering code with radius $R$. An upper bound of the smallest array…
Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…
In combinatorics, a derangement is a permutation that has no fixed points. The number of derangements of an n-element set is called the n-th derangement number. In this paper, as natural companions to derangement numbers and degenerate…