Enumerator polynomial

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

MacWilliams identity

Denote the dual code of $C\subset \mathbb {F} _{2}^{n}$ by

$C^{\perp }=\{x\in \mathbb {F} _{2}^{n}\,\mid \,\langle x,c\rangle =0{\mbox{ }}\forall c\in C\}$

(where $<,>$ denotes the vector dot product and which is taken over $\mathbb {F} _{2}$).

The MacWilliams identity states that

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

The identity is named after Jessie MacWilliams.

References

• Hill, Raymond (1986). A first course in coding theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press. pp. 165–173. ISBN 0-19-853803-0.
• Pless, Vera (1982). Introduction to the theory of error-correcting codes. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons. pp. 103–119. ISBN 0-471-08684-3.