Coding Theory

Seventh Assignment, Reed-Solomon Codes: due Wed. May 9

1. Let F_16 = F_2[a] where a^4 + a + 1 = 0 . For each value of m from 2 to 15 find the minimum distance of the cyclic code over F_16 of length 15 generated by g(x) = (a + x)(a^m + x) .
2. Let F_32 = F_2[a] where a^5 + a^2 + 1 = 0 . For each value of m from 2 to 31 find the minimum distance of the cyclic code of length 31 over F_32 generated by g(x) = (a + x)(a^m + x) .
3. Do 6.1.7 for g(x) = (b^3 + x)(b^4 + x)(b^5 + x) (I'm using b for beta). Answer in addition:
   • (a') What are the dimension and minimum distance of C.
   • (c) Just do (i) and (ii).
   • (d') Find the dimension and the designed distance of the binary subfield subcode.
4. CT 6.2.8. In part (c) just do (i).
5. (See CT 6.2.9 and 10). Let F_32 = F_2[a] where a^5 + a^2 + 1 = 0 . Find a generator polynomial g(x) for the Reed-Solomon code with designed distance 7 such that m = 8 is the smallest power of a that is a root of g(x) . Find a generator polynomial for C^perp as well. Finally, express each of C and C^perp as the span of vectors of the form [1, a^i, a^(2i), ...,a^((n-1)i)]
6. Let F_16 be as defined above and let g(x) = (1 + x)(a + x)(a^2 + x)(a^3 + x). Using the decoding algorithm from Monday, find the error vector if the syndrome is
   1. s = [1, a^7, a^5, a^9]
   2. s= [a^4, a^12, a^5, a^13]
   3. s= [a^4, a^12, a^5, a^2]