Homework 6

                                        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 .

                                     p(x) = x⁴ + x + 1 .
 

Let  g(x)  be the binary polynomial of smallest degree having
 

                                     a  and  a⁵

as roots.  Let  V = ( g(x) )  be the cyclic code of smallest length having  g(x)  as a generator polynomial.  Use  a  and  a⁵  to construct a parity check matrix  H  of  V .  (Do not explicitly compute  g(x) .  )

Hint: Let  K  denote the matrix:

                [ 1    a      a²      a³      a⁴        ...     a¹⁴          ]
    K=       [                                                                         ]
                [ 1 (a⁵)¹ (a⁵)² (a⁵)³ (a⁵)⁴  ...   (a⁵)]¹⁴    ]

Then

                          f(x) = f₀+f₁x+f₂x²+ ... f₁₄x¹⁴   e   V

iff   

                    (f₀, f₁, f₂, ... , f₁₄)K^T = [ f(a), f(a⁵) ] = [ 0, 0 ]

Next  let  H'  denote the binary matrix formed by replacing each element of  GF(2⁴)  in  K'  with the corresponding binary 4-tuple written as a column vector.  Then the row space of  H'  is  V^Perp.  Unfortunately, the rows of  H'  may be linearly dependent.  Form the parity check matrix  H  by putting  H'  in row echelon canonical form and deleting the all zero rows.