Enumerator polynomial

<h2 id="basic-properties">Basic properties</h2>
<ol><li>
 
 
 
 W
 (
 C
 ;
 0
 ,
 1
 )
 =
 
 A
 
 0
 
 
 =
 1
 
 
 {\displaystyle W(C;0,1)=A_{0}=1}
 
</li>
<li>
 
 
 
 W
 (
 C
 ;
 1
 ,
 1
 )
 =
 
 ∑
 
 w
 =
 0
 
 
 n
 
 
 
 A
 
 w
 
 
 =
 
 |
 
 C
 
 |
 
 
 
 {\displaystyle W(C;1,1)=\sum _{w=0}^{n}A_{w}=|C|}
 
</li>
<li>
 
 
 
 W
 (
 C
 ;
 1
 ,
 0
 )
 =
 
 A
 
 n
 
 
 =
 1
 
 
  if 
 
 
 (
 1
 ,
 …
 ,
 1
 )
 ∈
 C
  
 
 
  and 
 
 
 0
 
 
  otherwise
 
 
 
 
 {\displaystyle W(C;1,0)=A_{n}=1{\mbox{ if }}(1,\ldots ,1)\in C\ {\mbox{ and }}0{\mbox{ otherwise}}}
 
</li>
<li>
 
 
 
 W
 (
 C
 ;
 1
 ,
 −
 1
 )
 =
 
 ∑
 
 w
 =
 0
 
 
 n
 
 
 
 A
 
 w
 
 
 (
 −
 1
 
 )
 
 n
 −
 w
 
 
 =
 
 A
 
 n
 
 
 +
 (
 −
 1
 
 )
 
 1
 
 
 
 A
 
 n
 −
 1
 
 
 +
 …
 +
 (
 −
 1
 
 )
 
 n
 −
 1
 
 
 
 A
 
 1
 
 
 +
 (
 −
 1
 
 )
 
 n
 
 
 
 A
 
 0
 
 
 
 
 {\displaystyle 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}}
 
</li></ol>
<h2 id="macwilliams-identity">MacWilliams identity</h2>
Denote the <a href="/facts/Dual_code/AK5TS6nA">dual code</a> of 
 
 
 
 C
 ⊂
 
 
 F
 
 
 2
 
 
 n
 
 
 
 
 {\displaystyle C\subset \mathbb {F} _{2}^{n}}
 
 by

C
          
            ⊥
          
        
        =
        {
        x
        ∈
        
          
            F
          
          
            2
          
          
            n
          
        
        
        ∣
        
        ⟨
        x
        ,
        c
        ⟩
        =
        0
        
          
             
          
        
        ∀
        c
        ∈
        C
        }
      
    
    {\displaystyle C^{\perp }=\{x\in \mathbb {F} _{2}^{n}\,\mid \,\langle x,c\rangle =0{\mbox{  }}\forall c\in C\}}

(where 
 
 
 
 ⟨
  
 ,
  
 ⟩
 
 
 {\displaystyle \langle \ ,\ \rangle }
 
 denotes the vector <a href="/facts/Dot_product/tNz8MLjT">dot product</a> and which is taken over 
 
 
 
 
 
 F
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {F} _{2}}
 
).
The MacWilliams identity states that

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

The identity is named after <a href="/facts/Jessie_MacWilliams/pGWBIGmI">Jessie MacWilliams</a>.

<h2 id="distance-enumerator">Distance enumerator</h2>
The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers

A
          
            i
          
        
        =
        
          
            1
            M
          
        
        #
        
          {
          
            (
            
              c
              
                1
              
            
            ,
            
              c
              
                2
              
            
            )
            ∈
            C
            ×
            C
            ∣
            d
            (
            
              c
              
                1
              
            
            ,
            
              c
              
                2
              
            
            )
            =
            i
          
          }
        
      
    
    {\displaystyle 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
        )
        =
        
          ∑
          
            i
            =
            0
          
          
            n
          
        
        
          A
          
            i
          
        
        
          x
          
            i
          
        
        
          y
          
            n
            −
            i
          
        
      
    
    {\displaystyle 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
          
        
        =
        #
        
          {
          
            c
            ∈
            C
            ∣
            d
            (
            c
            ,
            x
            )
            =
            i
          
          }
        
        .
      
    
    {\displaystyle 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,...,An).
A code C is regular if the rows of B corresponding to the codewords of C are all equal.

<ul><li>Hill, Raymond (1986). <a href="https://archive.org/details/firstcourseincod0000hill">A first course in coding theory</a>. Oxford Applied Mathematics and Computing Science Series. <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>. pp. <a href="https://archive.org/details/firstcourseincod0000hill/page/165">165–173</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-853803-0.</li>
<li><a href="/facts/Vera_Pless/na9l860I">Pless, Vera</a> (1982). <a href="/facts/Introduction_to_the_Theory_of_Error-Correcting_Codes/VKWbax0T">Introduction to the theory of error-correcting codes</a>. Wiley-Interscience Series in Discrete Mathematics. <a href="/facts/John_Wiley_%26_Sons/53g3LqJc">John Wiley & Sons</a>. pp. 103–119. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-08684-3.</li>
<li>J.H. van Lint (1992). <a href="https://archive.org/details/introductiontoco0000lint">Introduction to Coding Theory</a>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 86 (2nd ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-54894-7. Chapters 3.5 and 4.3.</li></ul>

Enumerator polynomial open-in-new

Enumerator polynomial