A note on optimum burst-error-correcting codes

A detailed study has been made of a certain class of systematic binary error-correcting codes that will correct the error bursts typical of some digital channels. These codes--generalizations of codes discovered by Abramson and Melas--are cyclic codes designed to correct any single burst of errors per
n
-digit word provided that the width of the burst (regarded cyclically) does not exceed a certain number of digits,
b
. Moreover, these codes are optimum in the sense that they employ the minimum number of redundant digits theoretically possible for a cyclic code with given values of
n
and
b
. A cyclic code is completely characterized by its generator polynomial
g(x)
, hence, the properties of the code can be determined by analysis of the corresponding
g(x)
. Necessary and sufficient conditions on
g(x)
have been formulated for the corresponding cyclic code to be an optimum burst-
b
correcting code. These conditions have been formulated into a series of tests that can be carried out (in principle) on any
g(x)
. All optimum burst-
b
cyclic codes with
n < 2^{12}
and
b < 6
have been found in this way and their generators are tabulated in the paper. In all, 98 codes are listed (not counting reciprocals) for
b = 3
and
b = 4
; it was shown that no optimum codes exist for
b = 5
within the limits stated. Practical codes for
b \geq 6
will probably be nonoptimum codes because of the extreme word lengths required for optimum ones.