- MacWilliams & Sloane, Chap. 7
- Peterson & Weldon, Chap. 8
- Berlekamp, Chap. 5
Compute the addition and multiplication tables for the ring
R3 = GF(2)[x]/(x3+1)
Also express each of the following ideals in the ring R3 as a set of elements of R3 .
(0), (1+x), (x2+x+1), (1), (x2+1), (x3+1), (x5+x+1).
For example,
(0) = { 0 } and ( x4 + x2 +1 ) = { 0, x2 + x+1 }
Let V be a binary linear code given by the generator matrix
[ 101011 ] G = [ 011110 ] [ 000111 ]
- a) Find a parity check matrix H of V .
- b) Construct a maximum likelihood decoding
table for V .
- c) Use H to reduce the maximum likelihood
decoding table of b) to an error/syndrome table.
- d) Demonstrate how your error/syndrome table can be used to decode
the
received vector r = 111101.
- e) Use the generator matrix to create a list of all code vectors of V. Then use this list to determine the minimum distance of V.
Devise a procedure for correcting double erasures for the Hamming [2m-1, 2m-1-m, 3] code.