NDLI: Well-Spaced Labelings of Points in Rectangular Grids

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Well-Spaced Labelings of Points in Rectangular Grids

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lagarias, Jeffrey C.
Copyright Year	2000
Abstract	We describe methods to label the M1 M2 grid with the integers 1 to M1M2 so that any K consecutively labeled cells are relatively far apart in the grid in the Manhattan metric. Constructions of such labelings are given which are nearly optimal in a range of conditions. Such labelings can be used in addressing schemes for storing data on two-dimensional arrays that include randomly located "blobs" of defective cells. The data can be precoded using block error-correcting codes before storage, and the usefulness of well-spaced points is to decrease the probability of "burst" errors which cannot be corrected. Possible applications include the storage of speech or music on low-quality memory chips and in "holographic memories" to store bit-mapped data.More generally, we present a general family of mappings of the integers 1 to M1M2 . . . Md onto the d-dimensional grid of size M1 M2 . . . Md, called mixed radix vector mappings. These mappings give labelings whenever they are one to one. We give a sufficient condition for these mappings to be one to one, which is easy to verify in many cases.
Starting Page	521
Ending Page	534
Page Count	14
File Format	PDF
ISSN	08954801
DOI	10.1137/S0895480199362241
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	13
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-01
Access Restriction	Subscribed
Subject Keyword	Relations with coding theory Packing and covering Tilings in $2$ dimensions Other types of codes Combinatorics Combinatorial optimization radix expansions error correction packings Radix representation; digital problems
Content Type	Text
Resource Type	Article
Subject	Mathematics

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