Coding with permutations

International Journal of Information and Coding Theory, 2010

We look at some techniques for constructing permutation arrays using projections in finite projective spaces and the geometry of arcs in the finite projective plane. We say a permutation array P A(n, d) has length n and minimum distance d when it consists of a collection of permutations on n symbols that pairwise agree in at most n − d coordinate positions. Such arrays can also be viewed as non-linear codes and are used in powerline communication. While our techniques likely do not produce optimal arrays, we are able to construct examples of codes for certain parameter sets where no known constructions were previously known.

IEEE Transactions on Information Theory, 2000

Permutation arrays have found applications in powerline communication. One construction method for permutation arrays is to map good codes to permutations using a distance-preserving mappings (DPM). DPMs are mappings from the set of all q-ary vectors of a fixed length to the set of permutations of some fixed length (the same or longer) such that every two distinct vectors are mapped to permutations with the same or larger Hamming distance than that of the vectors. A DPM is called distance increasing (DIM) if the distances are strictly increased (except when the two vectors are equal). In this correspondence, we propose constructions of DPMs and DIMs from ternary vectors. The constructed DPMs and DIMs improve many lower bounds on the maximal size of permutation arrays.

IEEE Transactions on Information Theory, 2003

Mappings of the set of binary vectors of a fixed length to the set of permutations of the same length are useful for the construction of permutation codes. In this correspondence, several explicit constructions of such mappings preserving or increasing the Hamming distance are given. Some applications are given to illustrate the usefulness of the construction. In particular, a new lower bound on the maximal size of permutation arrays (PAs) is given.

2014 IEEE International Symposium on Information Theory, 2014

The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile memory storage. Error-correction in the rank modulation scheme is done by considering permutation codes. In this paper we consider codes in the set of all permutations on n elements, Sn, using the Kendall's τ-metric. We prove that there are no perfect single-error-correcting codes in Sn, where n > 4 is a prime or 4 ≤ n ≤ 10. We also prove that if such a code exists for n which is not a prime then the code should have some uniform structure. We define some variations of the Kendall's τ-metric and consider the related codes and specifically we prove the existence of a perfect single-error-correcting code in S5. Finally, we examine the existence problem of diameter perfect codes in Sn and obtain a new upper bound on the size of a code in Sn with even minimum Kendall's τ-distance.

Annales Mathematicae et Informaticae

Permutations of [ ] = {1, 2,. .. , } may be represented geometrically as bargraphs with column heights in [ ]. We define the notion of capacity of a permutation to be the amount of water that the corresponding bargraph would hold if the region above it could retain water assuming the usual rules of fluid flow. Let () be the sum of the capacities of all permutations of [ ]. We obtain, in a unique manner, all permutations of length +1 from those of length , which yields a recursion for (+ 1) in terms of () that we can subsequently solve. Finally, we consider permutations that have a single dam (i.e., a single area of water containment) and compute the total number and capacity of all such permutations of a given length. We also provide bijective proofs of these formulas and an asymptotic estimate is found for the average capacity as increases without bound.

Zeitschrift für Operations Research, 1986

This note deals with the problem of permuting elements within columns of a real matrix so as to minimize a real-valued function of row sums. The special case dealing with minimization of maximum row sum has been studied by several authors 6, recently. Here we are concerned primarily with the case in which the matrix has two columns only and the function is Schur-convex.

Electronic Notes in Discrete Mathematics, 2005

The X-ray of a permutation is defined as the sequence of antidiagonal sums in the associated permutation matrix. X-rays of permutation are interesting in the context of Discrete Tomography since many types of integral matrices can be written as linear combinations of permutation matrices. This paper is an invitation to the study of X-rays of permutations from a combinatorial point of view. We present connections between these objects and nondecreasing differences of permutations, zero-sum arrays, decomposable permutations, score sequences of tournaments, queens' problems and rooks' problems.

Discrete Mathematics, 2011

A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.

Journal of the London Mathematical Society, 1979

2007

A permutomino of size n is a polyomino whose vertices define a pair of distinct permutations of length n. In this paper we treat various classes of convex permutominoes, including the parallelogram, the directed convex and the stack ones. Using bijective techniques we provide enumeration for each of these classes according to the size, and characterize the permutations which are associated with permutominoes of each class.