Comparison of the 3-out-of-7 ARQ with Bose-Chaudhuri-Hocquenghem Coding Systems

The performance of coding schemes can be expressed by two criteria: the useful transmission rate
R
, and output character-error probability
P
. Both quantities are expressed in a form applicable to a binary forward channel coupled with an arbitrary feedback channel. Using these criteria the 3-out-of-7 ARQ code is compared with various error-correcting and detecting alternatives of the Bose-Chaudhuri-Hocquenghem codes of length
2^{m} - 1
, where
m = 3, 4, 5, 6, 7, 8
. It is found that for a specified binary element-error probability P
e
one can often find a Bose-ChaudhuriHocquenghem code which is considerably superior. If the element errors are independent of each other a purely error-correcting arrangement looks attractive. If the errors are correlated (i.e., occur in bursts) error detection holds more promise.
A (255, 187)
code, which is used to correct up to 8 errors and to detect others, shows a superiority which extends over all significant values of P
e
.