The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers
where i ranges from 0 to n. The distance enumerator polynomial is
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
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.
Referencesedit
Hill, Raymond (1986). A first course in coding theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press. pp. 165–173. ISBN0-19-853803-0.
enumerator, polynomial, coding, theory, weight, enumerator, polynomial, binary, linear, code, specifies, number, words, each, possible, hamming, weight, displaystyle, subset, mathbb, binary, linear, code, length, displaystyle, weight, distribution, sequence, n. In coding theory the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight Let C F 2 n displaystyle C subset mathbb F 2 n be a binary linear code 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 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 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 Contents 1 Basic properties 2 MacWilliams identity 3 Distance enumerator 4 ReferencesBasic properties editW C 0 1 A 0 1 displaystyle W C 0 1 A 0 1 nbsp W C 1 1 w 0 n A w C displaystyle W C 1 1 sum w 0 n A w C nbsp 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 nbsp 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 nbsp MacWilliams identity editDenote the dual code of C F 2 n displaystyle C subset mathbb F 2 n nbsp 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 nbsp where displaystyle langle rangle nbsp denotes the vector dot product and which is taken over F 2 displaystyle mathbb F 2 nbsp 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 nbsp The identity is named after Jessie MacWilliams Distance enumerator editThe 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 nbsp 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 nbsp 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 nbsp 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 References editHill 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 amp Sons pp 103 119 ISBN 0 471 08684 3 J H van Lint 1992 Introduction to Coding Theory GTM Vol 86 2nd ed Springer Verlag ISBN 3 540 54894 7 Chapters 3 5 and 4 3 Retrieved from https en wikipedia org w index php title Enumerator polynomial amp oldid 1181598138 Distance enumerator, wikipedia, wiki, book, books, library,