Reed-Solomon Code RS16(17, 19) - Number of Correctable and Detectable Symbols

Question

In Reed-Solomon coding with 16-bit symbols, for a configuration RS16(17, 19) where there are 17 data symbols, 2 parity symbols, and a total of 19 symbols, how many symbols are correctable and how many symbols are detectable?

correction capability t = (19 - 17) / 2 = 1 symbol

detection capability d = 19 - 17 + 1 = 3 symbol

is this correct?

Answer 1

correction = 1, detection = 2, distance = 3. There are some 3 error cases that appear to be valid codewords, as if no errors.

Assume the last 3 symbols are in error, and the first consecutive root is 1, then for a failure case, the two syndromes are:

s(0) =     e0 +     e1 +     e2 = 0
s(1) = 2^0 e0 + 2^1 e1 + 2^2 e2 = 0
s(1) =     e0 + 2   e1 + 4   e2 = 0

e0 + e1 + e2 = 0
e0 = e1 + e2
(e1 + e2) + 2 e1 + 4 e2 = 0
3 e1 + 5 e2 = 0

let e2 = any non-zero value
e1 = (5/3) e2
e0 = e1 + e2

Answer 2

The code has minimum distance $3 = 19-17+1$. The decoder can be designed to provide either guaranteed detection of all single or double errors (and many but by no means all triple errors) and no error-correction whatsoever, or guaranteed correction of all possible single errors, and detection of many but by no means all double errors.