Problems 6 (first draft)
1. Let C, C' be q-ary block length n codes.
Is their intersection a code?
Is their union a code?
What about the linear property?
What about the cyclic property?
2. Let C, C' be q-ary block length n linear codes.
(a) Show that their intersection is also such a code.
(b) Suppose that C, C' are also cyclic.
(i) Show that their intersection is cyclic.
Suppose C, C' have generator polynomials g, g' respectively.
What can you say about the generator for the intersection in
terms of these.
(ii) Show that C+C' (as defined in lectures) is a cyclic code.
What can you say about the generator for C+C' in terms of g and g'?
(Hint: consider our discussion of Euclid's algorithm in class.)
3. Construct a minimum distance 99 code.
Compare the rate of repetition codes and the G_23 Golay code.
4. Find all irreducible quadratics in F_2[x].
5. Find all irreducible cubics in F_2[x].
Consider the problem of finding irreducible quartics. What is new here?