Enumerator polynomial

In <a href="/facts/Coding_theory/SBqStoGL">coding theory</a>, the weight enumerator polynomial of a binary <a href="/facts/Linear_code/HoFI3eyF">linear code</a> specifies the number of words of each possible <a href="/facts/Hamming_weight/dJIW7qg4">Hamming weight</a>.
Let 
 
 
 
 C
 ⊂
 
 
 F
 
 
 2
 
 
 n
 
 
 
 
 {\displaystyle C\subset \mathbb {F} _{2}^{n}}
 
 be a binary linear code of length 
 
 
 
 n
 
 
 {\displaystyle n}
 
. The weight distribution is the sequence of numbers

A
          
            t
          
        
        =
        #
        {
        c
        ∈
        C
        ∣
        w
        (
        c
        )
        =
        t
        }
      
    
    {\displaystyle A_{t}=\#\{c\in C\mid w(c)=t\}}

giving the number of <a href="/facts/Code_word_(communication)/sASREopJ">codewords</a> c in C having weight t as t ranges from 0 to n. The weight enumerator is the bivariate <a href="/facts/Polynomial/Lzak8VVx">polynomial</a>

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

Enumerator polynomial open-in-new

Enumerator polynomial