# Math 342 Homework Assignment #5 (Due Thursday, March 24)

```Math 342
Homework Assignment #5 (Due Thursday, March 24)
1. For each linear code C, over GF (q), defined by the following generator matrices, find
a parity check matrix for C and generator and parity check matrices for C ⊥ .
(a) q = 3


1 0 0 0 1 2
G= 0 1 0 2 0 2 
0 0 1 2 0 0
(b) q = 4


0 1 0 1 a b
G= b 0 0 1 1 0 
a a a a b 1
2. Find the minimum distance of the linear codes over GF (q) defined by each of the
following parity check matrices.
(a) q = 3
H=
2 2 2 1
1 0 2 2
(b) q = 4


a b 1 1
H= b b a 0 
1 a a b
3. Let C be the linear code with parity check matrix


1 1 1 1 0 0
H= 0 1 1 0 1 0 
1 1 0 0 0 1
(a) Construct a syndrome decoding table, i.e., for each coset of C find a coset leader
and the corresponding syndrome.
(b) Decode each of the following received vectors
i. 110010
ii. 111111
4. Let C and D be linear codes in V (n, q).
Let C + D = {x + y : x ∈ C, y ∈ D}.
(a) Show that C + D is a linear code.
(b) Show that (C + D)⊥ = C ⊥ ∩ D⊥
(c) Show that (C ∩ D)⊥ = C ⊥ + D⊥
1
```