Homework 5

• Reading Assignment
  • MacWilliams & Sloane, Chap. 7
  • Peterson & Weldon, Chap. 8
  • Berlekamp, Chap. 5

Problem 1.  Let  V   denote the cyclic code of length  15   in  R₁₅  given by the generator polynomial:

                                  g(x) = x⁸ + x⁴ + x² + x + 1.
 

• (a) Use  g(x)  to compute the generator matrix  G  of  V.  What is the dimension of  V ?
• (b) Compute the parity check polynomial   h(x)  of  V.
• (c) Use  h(x)  to compute a generator matrix   of V^! .
• (d) From  h(x),  compute the generator polynomial  of  V^PERP.
• (e) Use the polynomial computed in (d)   to compute the parity check matrix  H   of  V.

---

Problem 2.  Let  V = ( g(x))   be the binary cyclic code of length  7  with generator polynomial

                                           g(x) = x⁴ + x² + x +1
 

• (a) Find the systematic basis  { x^j + r_j(x)}   of  V .
• (b) Use the above systematic basis of  V  to construct a systematic generator matrix  G of  V .
• (c) Use the remainders r_j(x)  to construct a systematic parity check matrix  H of  V .

Last Modified: November 1, 1999