Coding Theory
Seventh Assignment, Reed-Solomon Codes: due Wed.
May 9
- 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) .
- 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) .
- 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.
- CT 6.2.8. In part (c) just do (i).
- (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)]
- 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
- s = [1, a^7, a^5, a^9]
- s= [a^4, a^12, a^5, a^13]
- s= [a^4, a^12, a^5, a^2]