Suppose a generator polynomial gx for generating error detec

Suppose a generator polynomial g(x) for generating error detection code has even number of non-zero terms [for example, polynomials g(x) = X⁷+X⁴+X+1 and g(x )=X¹⁵+X¹¹+X⁸+X⁵+X³+1 have even number (four and six, respectively) of non-zero terms whereas polynomial g(x) =X¹⁵+X¹¹+X⁸+X³+1 has odd number (five) of non-zero terms]. Show that code words generated by such a polynomial have even parity. What is your assessment of parity of codewords generated by a generator polynomial with odd number of non-zero terms.

Solution

A generating polynomial with even number of non-zero term has even number of 1-bit in its code word. Such as,

g(x)=X7+X4+X+1 has a code word 10010011 (even number of 1-bit)

Now any binary code with even number of 1-bit has even parity. S0, the code will have even parity if there is even number of non-zero term in generating polynomial.

Similarly for odd number of non-zero term in generating polynomial will give a odd number of 1-bit in code word. Which will give a odd parity.