Frequency permutation arrays

Abstract

Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n ¼ m and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency , such that the Hamming distance between any distinct x; y 2 T is at least d. Such arrays have potential applications in powerline communication. In this article, we establish basic properties of FPAs, and provide direct constructions for FPAs using a range of combinatorial objects, including polynomials over finite fields, combinatorial designs, and codes. We also provide recursive constructions, and give bounds for the maximum size of such arrays.

Equidistant frequency permutation arrays and related constant composition codes

“…, m − 1, the code corresponds to a PA: a PA of length n, minimum distance d and size v, defined on the elements of an n-set S, is a v × n array such that each row is a permutation of the symbols of S and any two rows agree in at most n − d columns, or precisely n − d columns for an EPA. More generally, the CCCs in which µ 0 = · · · = µ m−1 = µ for n = mµ correspond to FPAs ( [10]). A FPA of length n = mµ, frequency µ, distance d and size v, defined on the elements of a m-set S, is a v × n array in which each row is a multipermutation of the symbols of S each repeated µ times and any two rows agree in at most n − d columns, or precisely n − d columns for an equidistant FPA (EFPA).…”

Multipermutation Codes in the Ulam Metric for Nonvolatile Memories

“…For the case of the Hamming distance, we find the asymptotic capacity, while for the Ulam distance we provide lower and upper bounds on the capacity. We point out that a number of bounds on multipermutation codes in the Hamming metric were derived in [15], including some simple and some complicated expressions involving Laguerre polynomials. Nevertheless, these bounds do not allow for finding a capacity formula for the underlying codes.…”

“…In the case of the Poisson transform, provided that some minor technical conditions are met, it can be shown that f (m, l) ≃ g(m, l/m), where a(x) ≃ b(x) stands for lim x→ ∞ a(x)/b(x) = 1. Intuitively, the aforementioned result implies that when the dependencies among a large number of random variables are weak -for example, only in terms of a constraint on the total sum of their values -then the [15] 120960 120960 60480 20160 5040 1008 168 24 3 Lemma 4 19683 6561 2187 729 243 81 27 9 3 (11) (approximate bound) 12077 4560 1700 624 224 79 27 9 3 (9) 1680 1680 1050 510 210 78 27 9 3 variables are asymptotically independent, provided a proper choice of the distribution ensures consistence with the finitevalued parameters.…”

“…expression for the size of the ball may be found in [15]). The Lemma follows by a standard application of Gilbert's argument.…”

Constructions of optimal GDRP(n,λ;v)’s of type λ1Yan1, Yin2 2008Discrete Applied Mathematics8000View full textAdd to dashboardCite

“…The class of constant composition codes includes the important permutation codes and have attracted recent interest due to their numerous applications (see, for example, [2][3][4]6,10,17] and the references therein). Ding and Yin [7,8] gave a combinatorial characterization of constant composition codes.…”