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 $\displaystyle C_{1}$ and $\displaystyle C_{2}$. The code $\displaystyle C_{1}$ has length 7, distance 3 and has the 4 digit binary strings as the message words for the code. The code $\displaystyle C_{2}$ is the first-order Reed-Muller code $\displaystyle R(6)$. 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 $\displaystyle C_{2}$ codeword have all their symbls marked as erasures. Received words at distance less than *r* must be corrected by the $\displaystyle C_{2}$decoder and the number of errors which the $\displaystyle C_{2}$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

$\displaystyle R(6)$ decoder is less than 1 × $\displaystyle 10^-11$. Find *r*.

(b)

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).