Chapter

# Packing Arrays

DOI: 10.1007/3-540-45995-2_28
Source: DBLP

ABSTRACT

A packing array is a b×k array of values from a g-ary alphabet such that given any two columns, i and j, and for all ordered pairs of elements from the g-ary alphabet, (g1, g2), there is at most one row, r, such that ar,i = g1 and ar,j = g2. Further, there is a set of at least n rows that pairwise differ in each column: they are disjoint. A central question is to determine, for given g and k, the maximum possible b. We developg eneral direct and recursive constructions and upper bounds on the sizes of packing arrays. We also show the
equivalence of the problem to a matching problem on graphs and a class of resolvable pairwise balanced designs. We provide
tables of the best known upper and lower bounds.

### Full-text

Available from: Eric Mendelsohn, Aug 04, 2014
• ##### Article: Packing Arrays and Packing Designs
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ABSTRACT: A packing array is a b × k array, A with entriesa i,j from a g-ary alphabet such that given any two columns,i and j, and for all ordered pairs of elements from a g-ary alphabet,(g 1, g 2), there is at most one row, r, such thata r,i = g 1 anda r,j = g 2. Further, there is a set of at leastn rows that pairwise differ in each column: they are disjoint. A central question is to determine, forgiven g, k and n, the maximum possible b. We examine the implications whenn is close to g. We give a brief analysis of the case n = g and showthat 2g rows is always achievable whenever more than g exist. We give an upper bound derivedfrom design packing numbers when n = g − 1. When g + 1 ≤k then this bound is always at least as good as the modified Plotkin bound of [12]. When theassociated packing has as many points as blocks and has reasonably uniform replication numbers, we show thatthis bound is tight. In particular, finite geometries imply the existence of a family of optimal or near optimalpacking arrays. When no projective plane exists we present similarly strong results. This article completelydetermines the packing numbers, D(v, k, 1), when $$v < \frac{{k(k - 1)}}{2}$$.
No preview · Article · Jan 2002 · Designs Codes and Cryptography
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##### Article: Class-Uniformly Resolvable Group Divisible Structures I: Resolvable Group Divisible Designs
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ABSTRACT: We consider Class-Uniformly Resolvable Group Divisible Designs (CURGDD), which are resolvable group divisible designs in which each of the resolution classes has the same number of blocks of each size. We derive the fully general necessary conditions including a number of extremal bounds. We present some general constructions including a novel construction for shrinking the index of a master design. We construct a number of in nite families, primarily with block sizes 2 and k, including some extremal cases.
Preview · Article · Apr 2004 · The electronic journal of combinatorics
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##### Article: On constant composition codes
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ABSTRACT: A constant composition code over a k-ary alphabet has the property that the numbers of occurrences of the k symbols within a codeword is the same for each codeword. These specialize to constant weight codes in the binary case, and permutation codes in the case that each symbol occurs exactly once. Constant composition codes arise in powerline communication and balanced scheduling, and are used in the construction of permutation codes. In this paper, direct and recursive methods are developed for the construction of constant composition codes.
Preview · Article · Apr 2006 · Journal of Combinatorial Mathematics and Combinatorial Computing