the macwilliams identity + more Flashcards

1
Q

the macwilliams identity:

A

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

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2
Q

the weight enumerator of the trivial code:

A

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

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3
Q

the weight enumerator of En:

A

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

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4
Q

the average weight equation:

A

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

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5
Q

simplex code:

A

Σ(r,q) is defined as Ham(r,q)⊥

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6
Q

properties of a simplex code:

A

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

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7
Q

the weight enumerator of Ham(r,2):

A

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

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8
Q

the plotkin bound:

A

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

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