In 2014, Steele, Lea, and Flood presented
{\sc SplitMix}, an object-oriented pseudorandom number generator
({\sc prng}) that is quite fast (9 64-bit arithmetic/logical operations
per 64 bits generated) and also \emph{splittable}. A conventional
{\sc prng} object provides a \emph{generate} method that returns one
pseudorandom value and updates the state of the {\sc prng}; a splittable
{\sc prng} object also has a second operation, \emph{split}, that replaces
the original {\sc prng} object with two (seemingly) independent {\sc prng}
objects, by creating and returning a new such object and updating
the state of the original object. Splittable {\sc prng} objects make it
easy to organize the use of pseudorandom numbers in multithreaded
programs structured using fork-join parallelism.
This overall strategy still appears to be sound,
but the specific arithmetic calculation used for \emph{generate} in the {\sc SplitMix}
algorithm has some detectable weaknesses, and the period of any one generator
is limited to $2^{64}$.

Here we present the LXM \emph{family} of {\sc prng} algorithms.
The idea is an old one: combine the outputs of two independent {\sc prng} algorithms,
then (optionally) feed the result to a mixing function.
An LXM algorithm uses a linear congruential subgenerator and an $\mathbf{F}_2$-linear subgenerator;
the examples studied in this paper use a linear congruential generator (LCG) of period $2^{16}$, $2^{32}$, $2^{64}$, or $2^{128}$
with one of the multipliers recommended by L'Ecuyer
or by Steele and Vigna,
and an $\mathbf{F}_2$-linear xor-based generator (XBG) of the \texttt{xoshiro} family or \texttt{xoroshiro} family
as described by Blackman and Vigna.
For mixing functions we study the MurmurHash3 finalizer function;
variants by David Stafford, Doug Lea, and \texttt{degski}; and the null (identity) mixing function.

Like {\sc SplitMix}, LXM provides both a \emph{generate}
operation and a \emph{split} operation.
Also like {\sc SplitMix}, LXM requires no locking or other synchronization
(other than the usual memory fence after instance initialization),
and is suitable for use with {\sc simd} instruction sets
because it has no branches or loops.

We analyze the period and equidistribution
properties of LXM generators, and present the results of thorough testing of specific members of this family,
using the TestU01 and PractRand test suites, not only on single instances of the algorithm but also
for collections of instances, used in parallel, ranging in size from $2$ to $2^{24}$.
Single instances of LXM that include a strong mixing function
appear to have no major weaknesses, and LXM is significantly more robust than
{\sc SplitMix} against accidental correlation in a multithreaded setting.
We believe that LXM, like {\sc SplitMix}, is suitable for
``everyday'' scientific and machine-learning applications (but not cryptographic applications),
especially when concurrent threads or distributed processes are involved.