65.154 LowerBoundCoveringRadiusEmbedded1

LowerBoundCoveringRadiusEmbedded1( n, M, false )

LowerBoundCoveringRadiusEmbedded1( n, r [, true ] )

If the last argument of LowerBoundCoveringRadiusEmbedded1 is false, then it returns a lower bound for the covering radius of a code of size M and length n. Otherwise, it returns a lower bound for the size of a code of length n and covering radius r.

This bound only works for binary codes.

It is based on the following inequality:

M left( V_2(n,r) - binomial2rr right) geq 2^n - A( n, 2r+1 ) binomial2rr,

where A(n,d) denotes the maximal cardinality of a (binary) code of length n and minimum distance d. The function UpperBound is used to compute this value.

Sometimes LowerBoundCoveringRadiusEmbedded1 is better than
LowerBoundCoveringRadiusEmbedded2, sometimes it is the other way around.

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