Matrix approach to synchronization recovery for binary cyclic codes

The problem of detecting and correcting loss of synchronization for binary cyclic codes is examined. The method used is to form a coset code from tho given
(n, k)
cyclic code by adding a suitable
n
vector to each cyclic code word before transmission. Unlike some other techniques, the length of the code word
n
is not altered. A code word may be represented in at least two ways: 1) as a vector and 2) as a polynomial. Depending on the problem at hand, one approach may offer more insight or supply a shorter proof for a theorem than the other. Throughout this paper, we use the binary
n
-vector representation. Using this approach, we give new proofs to some known results, and derive some now theorems dealing with the simultaneous occurrence of loss of synchronization and additive errors. Specifically, it is shown that there exist coset codes that can correct both additive error and synchronization error even when they occur simultaneously.