**Problem 1.**

Form a maximum-likelihood decoding table fot the binary code consisting of the four code words 0000, 0011, 1100, and 1111,**a) Assuming the binary symmetric channel, b) Assuming the binary erasure channel.****Problem 2.**

A "metric" function is defined as a real-valued function having the following three properties:

Show that the Hamming distance is a metric function.**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)****Hint.**H(x,y) = Wt( x (+) y ), where H(x,y) denotes the Hamming distance between x and y, where Wt(v) denotes the Hamming weight of v (i.e., the number of 1's in v, and where (+) denotes componentwise addition mod 2.**Problem 3**

Show that the minimum Hamming distance at least 2t+1 between code blocks is necessary and sufficient for correcting all combinations of t or fewer errors.**Hint.**Use the triangle inequality.**Problem 4.**

Consider the kxn matrix

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 V**G = [ I, P ]**^{PERP}denote the vector space over F which is the orthogonal complement of V , i.e.,

where v**V**^{PERP}= { v in F^{n}| v^{.}u = 0 for all u in V },^{.}u denotes the inner product of vectors u and v . Prove that V^{PERP}is the row span of the matrix

where, in this case, I denotes the (n-k) x (n-k) identity matrix.**H = [ -P**^{Transpose}, I ],**Hint:**See Chapter 2, Section 6 of Peterson & Weldon.

Last Modified: September 25, 1997