an error correction encoding/decoding system for storing 4 digit binary sequences is set up using a 5-frame delay cross-interleave of two codes and . The code has length 7, distance 3 and has the 4 digit binary strings as the message words for the code. The code is the first-order Reed-Muller code . The data is stored as a sequence of binary digits on the tape with no alteration (eg modulation) other than the error correction encoding.
(a) The system must be designed so that for some interger r, received words which are distance r or greater from a codeword have all their symbls marked as erasures. Received words at distance less than r must be corrected by the decoder and the number of errors which the decoder corrects must be maximised. However, it is also required that the chance of a random received word (which is a random sequence of 64 binary digits as a result of a burst error) being
undetected by the
" /> decoder is less than 1 × 10^-11
" />. Find r.
Determine the smallest integer t such that there is a burst error of length t that the system cannot correct (in the presence of no other errors).