Enumerator polynomial

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

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

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

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

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

Basic properties

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

MacWilliams identity

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

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

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

The MacWilliams identity states that

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

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

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

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

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

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

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

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

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

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.
• J.H. van Lint (1992). Introduction to Coding Theory. GTM. 86 (2nd ed.). Springer-Verlag. ISBN 3-540-54894-7. Chapters 3.5 and 4.3.