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The Diametric Theorem in Hamming Spaces—Optimal Anticodes

✍ Scribed by Rudolf Ahlswede; Levon H. Khachatrian

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
222 KB
Volume
20
Category
Article
ISSN
0196-8858

No coin nor oath required. For personal study only.

✦ Synopsis

y g X X counts the number of different components,

we determine the maximal cardinality of subsets with a prescribed diameter d or, in another language, anticodes with distance d. We refer to the result as the diametric theorem.

In a sense anticodes are dual to codes, which have a prescribed lower bound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly.

Ž . We find it remarkable that the diametric theorem for arbitrary ␣ can be derived from our recent complete intersection theorem, which can be viewed as a Ž . diametric theorem for ␣ s 2 in the restricted case, where all n-length words considered have exactly k ones. ᮊ 1998 Academic Press 1. PREVIOUS RESULTS, CONJECTURES, AND THE NEW THEOREM w x This paper is another demonstration of the power of the methods of 3 . We stick to the earlier notation as far as possible and first repeat it. Then we state the complete intersection theorem in its historical context, because this enables us to put the new result into proper perspective. Here we need some more terminology for the formulation of known results and conjectures for the diametric problem in Hamming space or related intersection problems. Finally, we state the new diametric theorem.

‫ގ‬ denotes the set of positive integers and for i, j g ‫,ގ‬ ij, the set Ä 4 w x w x w x i, iq1, . . . , j is abbreviated as i, j . Moreover, for 1, j we also write j .

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