65.135 SphereContent

SphereContent( n, t, F )

SphereContent returns the content of a ball of radius t around an arbitrary element of the vectorspace <F>^<n>. This is the cardinality of the set of all elements of <F>^<n> that are at distance (see DistanceCodeword) less than or equal to t from an element of <F>^<n>.

In the context of codes, the function is used to determine if a code is perfect. A code is perfect if spheres of radius t around all codewords contain exactly the whole vectorspace, where t is the number of errors the code can correct.

    gap> SphereContent( 15, 0, GF(2) );
    1    # Only one word with distance 0, which is the word itself
    gap> SphereContent( 11, 3, GF(4) );
    4984
    gap> C := HammingCode(5);
    a linear [31,26,3]1 Hamming (5,2) code over GF(2)
    #the minimum distance is 3, so the code can correct one error
    gap> ( SphereContent( 31, 1, GF(2) ) * Size(C) ) = 2 ^ 31;
    true 

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