Normal-form game

In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight.

Let $C\subset \mathbb {F} _{2}^{n}$ be a binary linear code length $n$. The weight distribution is the sequence of numbers

$A_{t}=\#\{c\in C\mid w(c)=t\}$

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

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

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.

Distance enumerator

The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers

$A_{i}={\frac {1}{M}}\#\left\lbrace (c_{1},c_{2})\in C\times C\mid d(c_{1},c_{2})=i\right\rbrace$

where i ranges from 0 to n. The distance enumerator polynomial is

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

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

$B_{x,i}=\#\left\lbrace c\in C\mid d(c,x)=i\right\rbrace .$

The sum of the rows of B is M times the inner distribution vector (A₀,...,A_n).

A code C is regular if the rows of B corresponding to the codewords of C are all equal.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapters 3.5 and 4.3.