{{short description|Specifies the number of words of a binary linear code of each possible Hamming weight}} In coding theory, the '''weight enumerator polynomial''' of a binary linear code specifies the number of words of each possible Hamming weight.
Let <math>C \subset \mathbb{F}_2^n</math> be a binary linear code of length <math>n</math>. The '''weight distribution''' is the sequence of numbers
:<math> A_t = \#\{c \in C \mid w(c) = t \} </math>
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
:<math> W(C;x,y) = \sum_{w=0}^n A_w x^w y^{n-w}.</math>
==Basic properties== #<math> W(C;0,1) = A_{0}=1 </math> #<math> W(C;1,1) = \sum_{w=0}^{n}A_{w}=|C| </math> #<math> W(C;1,0) = A_{n}= 1 \mbox{ if } (1,\ldots,1)\in C\ \mbox{ and } 0 \mbox{ otherwise} </math> #<math> 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} </math>
==MacWilliams identity== Denote the dual code of <math>C \subset \mathbb{F}_2^n</math> by
:<math>C^\perp = \{x \in \mathbb{F}_2^n \,\mid\, \langle x,c\rangle = 0 \mbox{ }\forall c \in C \} </math>
(where <math>\langle\ ,\ \rangle</math> denotes the vector dot product and which is taken over <math>\mathbb{F}_2</math>).
The '''MacWilliams identity''' states that
:<math>W(C^\perp;x,y) = \frac{1}{\mid C \mid} W(C;y-x,y+x). </math>
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
:<math> A_i = \frac{1}{M} \# \left\lbrace (c_1,c_2) \in C \times C \mid d(c_1,c_2) = i \right\rbrace </math>
where ''i'' ranges from 0 to ''n''. The '''distance enumerator polynomial''' is
:<math> A(C;x,y) = \sum_{i=0}^n A_i x^i y^{n-i} </math>
and when ''C'' is linear this is equal to the weight enumerator.
The '''outer distribution''' of ''C'' is the 2<sup>''n''</sup>-by-''n''+1 matrix ''B'' with rows indexed by elements of GF(2)<sup>''n''</sup> and columns indexed by integers 0...''n'', and entries
:<math> B_{x,i} = \# \left\lbrace c \in C \mid d(c,x) = i \right\rbrace . </math>
The sum of the rows of ''B'' is ''M'' times the inner distribution vector (''A''<sub>0</sub>,...,''A''<sub>''n''</sub>).
A code ''C'' is '''regular''' if the rows of ''B'' corresponding to the codewords of ''C'' are all equal.
==References== * {{cite book | last=Hill | first=Raymond | title=A first course in coding theory | url=https://archive.org/details/firstcourseincod0000hill | url-access=registration | publisher=Oxford University Press | series=Oxford Applied Mathematics and Computing Science Series | date=1986 | isbn=0-19-853803-0 | pages=[https://archive.org/details/firstcourseincod0000hill/page/165 165–173] }} * {{cite book | last = Pless | first = Vera | authorlink=Vera Pless | title = Introduction to the theory of error-correcting codes|title-link= Introduction to the Theory of Error-Correcting Codes | publisher = John Wiley & Sons|series = Wiley-Interscience Series in Discrete Mathematics | date = 1982| isbn = 0-471-08684-3 | pages=103–119 }} * {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd | publisher=Springer-Verlag | series=GTM | volume=86 | date=1992 | isbn=3-540-54894-7 | url-access=registration | url=https://archive.org/details/introductiontoco0000lint }} Chapters 3.5 and 4.3.
Category:Coding theory Category:Error detection and correction Category:Mathematical identities Category:Polynomials