Math 342
Homework Assignment #4 (Due Thursday, March 10)
1. Show that row equivalence preserves linear independence (i.e., if A and B are row
equivalent, then the rows of A are linearly independent iff the rows of B are linearly
independent).
2. (We did the following problem in class for the special case p = 2).
Let p be prime and H be a subset of V (n, p). Show that the following are equivalent
(a) H is closed under vector addition
(b) H is a subgroup of V (n, p).
(c) H is a subspace of V (n, p).
3. Consider the matrix


aa1b
G =  0ba1 
a10a
(a) Verify that G is a generator matrix for a linear code C over GF (4).
(b) What is the dimension and what is the size of C?
(c) Encode each of the following message vectors to a codeword of C.
i. a1b
ii. a01
4. Find a generator matrix in standard form for the linear span (or an equivalent code)
of each of the following sets.
(a) {100111, 010110, 101010, 111100, 011011, 001101}, q = 2.
(b) {11212, 10110, 12011, 11202, 22121}, q = 3.
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