the macwilliams identity + more

Question 1

the macwilliams identity:

Answer 1

if C is a q-ary linear code, WC⊥(x,y) = (1/#C)WC(x+(q-1)y,x-y)

Question 2

the weight enumerator of the trivial code:

Answer 2

WF(x,y) = (1/#null)Wnull(x+(q-1)y,x-y) = (x+(q-1)y)^n

Question 3

the weight enumerator of En:

Answer 3

WE(x,y) = (1/Rep(n,F2))WRep(x+y,x-y)=(1/2)((x+y)^(n)+(x-y)^(n))

Question 4

the average weight equation:

Answer 4

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

Question 5

simplex code:

Answer 5

Σ(r,q) is defined as Ham(r,q)⊥

Question 6

properties of a simplex code:

Answer 6

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

Question 7

the weight enumerator of Ham(r,2):

Answer 7

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

Question 8

the plotkin bound:

Answer 8

if C in F is a binary linear code such that d=d(C)>(n/2), then #C<=d/(d-(n/2))