where is the -weight of the string .
If is an abelian group, one defines the Hamming weight enumerator of , denoted by , as a polynomial in only two indeterminates and :
that is one distinguishes only between zero and the non-zero letters of the strings in .
If is a code of block length , then both and are http://planetmath.org/node/6577homogeneous of degree . Therefore, one can set in in this case without losing information. The resulting polynomial can be uniquely rewritten in the form
the sequence defining the Hamming weight distribution. Analogously, one can define more general weight distributions by setting all but one indeterminate in equal to one.
Let be the ternary (that is ) linear code of block length the vectors , and . Then
and the Hamming weight distribution is .
The Hamming weight enumerator of the full binary code of length , , is simply given by , and the Hamming weight distribution is the -th of Pascal’s triangle.
|Date of creation||2013-03-22 15:13:23|
|Last modified on||2013-03-22 15:13:23|
|Last modified by||GrafZahl (9234)|
|Synonym||Hamming weight enumerator|
|Defines||complete weight enumerator|
|Defines||Hamming weight distribution|