In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight.
Let
C\subset
| n | |
| F | |
| 2 |
n
At=\#\{c\inC\midw(c)=t\}
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
W(C;x,y)=
| n | |
| \sum | |
| w=0 |
Awxwyn-w.
W(C;0,1)=A0=1
W(C;1,1)=
| n | |
| \sum | |
| w=0 |
Aw=|C|
W(C;1,0)=An=1if(1,\ldots,1)\inC and0otherwise
W(C;1,-1)=
| n | |
| \sum | |
| w=0 |
Aw(-1)n-w=An+(-1)1An-1+\ldots+(-1)n-1A1+(-1)nA0
Denote the dual code of
C\subset
| n | |
| F | |
| 2 |
C\perp=\{x\in
| n | |
| F | |
| 2 |
\mid\langlex,c\rangle=0\forallc\inC\}
(where
\langle , \rangle
F2
The MacWilliams identity states that
W(C\perp;x,y)=
| 1 | |
| \midC\mid |
W(C;y-x,y+x).
The identity is named after Jessie MacWilliams.
The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers
Ai=
| 1 | |
| M |
\#\left\lbrace(c1,c2)\inC x C\midd(c1,c2)=i\right\rbrace
where i ranges from 0 to n. The distance enumerator polynomial is
A(C;x,y)=
| n | |
| \sum | |
| i=0 |
Aixiyn-i
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
Bx,i=\#\left\lbracec\inC\midd(c,x)=i\right\rbrace.
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.
. Vera Pless . Introduction to the theory of error-correcting codes. Introduction to the Theory of Error-Correcting Codes . John Wiley & Sons. Wiley-Interscience Series in Discrete Mathematics . 1982. 0-471-08684-3 . 103–119 .