Mathematics 502
Homework (due Apr 4)
A. Hulpke
31) Let g(x) = 1 + x + x3 be a polynomial with coefficients in F2 .
a) Show that g(x) is a factor of x7 − 1 in F2 [x].
b) The polynomial g(x) is the generating polynomial for a cyclic code C . Find the generating matrix for C . c) Find a parity check matrix H for C . d) Show that for


1 1 0 1 0 0 0
 0 1 1 0 1 0 0 

G0 = 
 0 0 1 1 0 1 0 
0 1 1 1 0 0 1
we have that G0 H = 0.
e) Show that a permutation of the columns of G0 gives the generating matrix for a Hamming code.
32) Let g(x) be the generating polynomial for a cyclic code C of length n and h(x) =
xn − 1/ g(x) = b0 + b1 x + · · · + bl−1 xl−1 + xl . Show that the dual code C ⊥ is cyclic with
generating polynomial 1/b0 (1 + bl−1 x + · · · + b1 xl−1 + b0 xl ). (The factor 1/b0 is included
to make the highest nonzero coefficient be 1.)
33) Let C be a perfect linear e-error correcting code over Fq . Show that the weight enumerator of C is uniquely determined.
34) Let C be a binary perfect e-error correcting code of length n with 0 ∈ C . consider
the wiords of weight 2e + 1 in C as characteristic functions of 2e + 1-element subsets of
{1, . . . , n}. Prove that this is a (e + 1) − (n, 2e + 1, 1)-design.