a) Assuming the binary symmetric channel, b) Assuming the binary erasure channel.
a) d(x,y) = 0 iff x=y (Reflexivity) b) d(x,y) = d(y,x) (Symmetry) b) d(x,y) =< d(y,z) + d(x,z) (Triangle Inequality)Show that the Hamming distance is a metric function.
Hint.
Hint. Use the triangle inequality.
G = [ I, P ]over a field F, where I denotes the kxk identity matrix, and where P is a kx(n-k) matrix over F. Let V be the vector space over F spanned by the rows of G, and let VPERP denote the vector space over F which is the orthogonal complement of V , i.e.,
VPERP = { v in Fn | v.u = 0 for all u in V },where v.u denotes the inner product of vectors u and v . Prove that VPERP is the row span of the matrix
H = [ -PTranspose, I ],where, in this case, I denotes the (n-k) x (n-k) identity matrix.
Hint:
See Chapter 2, Section 6 of Peterson & Weldon.