We present an algorithm for a variant of the 3XOR problem with lists consisting of $n$-bit vectors whose coefficients are drawn independently at random according to a Bernoulli distribution of parameter $p < 1/2$. We show that in this particular context the problem can be solved much more efficiently than in the general setting. This leads to a linear algorithm with overwhelming success probability for $p < 0.0957$. With slightly different parameters, this method succeeds deterministically. The expected runtime is also linear for $p < 0.02155$ and always sub-quadratic.