Enumerator polynomial

Definition:

In mathematics, the weight enumerator of a binary linear code $C\subset \mathbb {F} _{2}^{n}$ of length $n$ is defined to be

$W(C;x,y)=\sum _{w=0}^{n}A_{w}x^{w}y^{n-w}$

where $A_{t}=\#\{c\in C\mid w(c)=t\}$ is defined to be the number of codewords $c\in C$ having Hamming weight $w(c)=t$.

Basic Properties:

1. $W(C;0,1)=A_{0}=1$

2. $W(C;1,1)=\sum _{w=0}^{n}A_{w}=|C|$

3. $W(C;1,0)=A_{n}=\mathbb {I} _{\{(1,\ldots ,1)\in C\}}\in \{0,1\}$

4. $W(C;1,-1)=\sum _{w=0}^{n}A_{w}(-1)^{n-w}=A_{n}-(-1)^{1}A_{n-1}+\ldots +(-1)^{n-1}A_{1}+(-1)^{n}A_{0}$

McWilliam's Identity:

Denote the dual code of $C\subset \mathbb {F} _{2}^{n}$ by $C^{\perp }=\{x\in \mathbb {F} _{2}^{n}{\mbox{ }}\mid {\mbox{ }}<x,c>=0{\mbox{ }}\forall c\in C\}$ (where $<,>$ denotes the vector dot product and which is taken over $\mathbb {F} _{2}$).

McWilliam's identity states:

$W(C^{\perp };x,y)={\frac {1}{\mid C\mid }}W(C;y-x,y+x)$