Enumerator polynomial
Encyclopedia
In coding theory
, the weight enumerator polynomial of a binary linear code
specifies the number of words of each possible Hamming weight
.
Let
be a binary linear code length 
. The weight distribution is the sequence of numbers
giving the number of codewords c in C having weight t as t ranges from 0 to n. The weight enumerator is the bivariate polynomial
by
(where
denotes the vector dot product
and which is taken over
).
The MacWilliams identity states that
The identity is named after Jessie MacWilliams
.
where i ranges from 0 to n. The distance enumerator polynomial is
and when C is linear this is equal to the weight enumerator.
The outer distribution of C is the 2n-by-n+1 matrix B with rows indexed by elements of GF(2)n and columns indexed by integers 0...n, and entries
The sum of the rows of B is M times the inner distribution vector (A0,...,An).
A code C is regular if the rows of B corresponding to the codewords of C are all equal.
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
, the weight enumerator polynomial of a binary linear code
Linear code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although Turbo codes can be seen as a hybrid of these two types. Linear codes allow for...
specifies the number of words of each possible Hamming weight
Hamming weight
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string...
.
Let
be a binary linear code length . The weight distribution is the sequence of numbersgiving the number of codewords c in C having weight t as t ranges from 0 to n. The weight enumerator is the bivariate polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
Basic properties
MacWilliams identity
Denote the dual code of by(where
denotes the vector dot productDot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
and which is taken over
).The MacWilliams identity states that
The identity is named after Jessie MacWilliams
Jessie MacWilliams
Florence Jessie MacWilliams was an English mathematician who contributed to the field of coding theory. She was born in Stoke-on-Trent, England and studied at the University of Cambridge, receiving her BA in 1938 and her MA in the following year. She moved to the United States in 1939 and studied...
.
Distance enumerator
The distance distribution or inner distribution of a code C of size M and length n is the sequence of numberswhere i ranges from 0 to n. The distance enumerator polynomial is
and when C is linear this is equal to the weight enumerator.
The outer distribution of C is the 2n-by-n+1 matrix B with rows indexed by elements of GF(2)n and columns indexed by integers 0...n, and entries
The sum of the rows of B is M times the inner distribution vector (A0,...,An).
A code C is regular if the rows of B corresponding to the codewords of C are all equal.