Shared-ACS Binary Code Pair Search Exploiting Coupled Symparity

Binary code negation and reversal are known to leave the autocorrelation sequence unchanged. However, for some code lengths, there exist code pairs for which the mechanism behind shared autocorrelation is not so simple; these are called shared-autocorrelation (or shared-ACS) binary code pairs. They are useful for, among other things, constructing binary and quad-phase Golay pairs of longer lengths. We define a symparity vector for any given binary code. For every shared-ACS binary code pairs up to length 39, the two code have the same symparity, or coupled symparity. Conditions are identified for which shared-ACS binary code pairs have coupled symparity. The coupled-symparity condition is used to develop an algorithm to find shared-ACS binary code pairs, with reduced complexity relative to more brute-force exhaustive searches. Although the algorithm assumes extra structure in the shared-ACS code pairs, it rediscovered all the sets of binary shared-ACS code pairs for lengths up to 20. It is then run for lengths 40 to 44, returning all the shared-ACS binary code pairs for each length. In particular, the algorithm results establish that there exist shared-ACS code pairs for the prime lengths 41 and 43.