  We consider random systems of equations x_1 + ... + x_k = a; 0 <= a <= 2
which are interpreted as equations modulo 3: We show for k >= 15 that the
satisfiability threshold of such systems occurs where the 2-core has density 1:
We show a similar result for random uniquely extendible constraints over 4
elements. Our results extend previous results of Dubois/Mandler for equations
mod 2 and k = 3 and Connamacher/Molloy for uniquely extendible constraints over
a domain of 4 elements with k = 3 arguments. Our proof technique is based on
variance calculations, using a technique introduced Dubois/Mandler. However,
several additional observations (of independent interest) are necessary.
