the macwilliams identity + more Flashcards
the macwilliams identity:
if C is a q-ary linear code, WC⊥(x,y) = (1/#C)WC(x+(q-1)y,x-y)
the weight enumerator of the trivial code:
WF(x,y) = (1/#null)Wnull(x+(q-1)y,x-y) = (x+(q-1)y)^n
the weight enumerator of En:
WE(x,y) = (1/Rep(n,F2))WRep(x+y,x-y)=(1/2)((x+y)^(n)+(x-y)^(n))
the average weight equation:
if C is a q-ary linear code of length n, the average of the weights of the codevectors of C is (n-z)(1-q^-1), where z is the number of zero columns in a generator matrix of C
simplex code:
Σ(r,q) is defined as Ham(r,q)⊥
properties of a simplex code:
the simpex code has length n=(q^(r)-1)/(q-1) and dimension r, the hamming distance between each pair of codevectors is q^r-1
the weight enumerator of Ham(r,2):
WHam(x,y) = (1/(n+1))((x+y)^(n)+n(x+y)^((n-1)/2)(x-y)^((n+1)/2)) where n=2^(r)-1
the plotkin bound:
if C in F is a binary linear code such that d=d(C)>(n/2), then #C<=d/(d-(n/2))