  Pareto optimal solutions are conceived for radar beamforming error (RBE) and
sum rate maximization in short-packet (SP) millimeter-wave (mmWave) integrated
sensing and communication (ISAC). Our ultimate goal is to realize
ultra-reliable low-latency communication (uRLLC) and real-time sensing
capabilities for 6G applications. The ISAC base station (BS) transmits short
packets in the downlink (DL) to serve multiple communication users (CUs) and
detect multiple radar targets (RTs). We investigate the performance trade-off
between the sensing and communication capabilities by optimizing both the radio
frequency (RF) and the baseband (BB) transmit precoder (TPC), together with the
block lengths. The optimization problem considers the minimum rate requirements
of the CUs, the maximum tolerable radar beamforming error (RBE) for the RTs,
the unit modulus (UM) elements of the RF TPC, and the finite transmit power as
the constraints for SP transmission. The resultant problem is highly non-convex
due to the intractable rate expression of the SP regime coupled with the
non-convex rate and UM constraints. To solve this problem, we propose an
innovative two-layer bisection search (TLBS) algorithm, wherein the RF and BB
TPCs are optimized in the inner layer, followed by the block length in the
outer layer. Furthermore, a pair of novel methods, namely a bisection
search-based majorizer and minimizer (BMM) as well as exact penalty-based
manifold optimization (EPMO) are harnessed for optimizing the RF TPC in the
inner layer. Subsequently, the BB TPC and the block length are derived via
second-order cone programming (SOCP) and mixed integer programming methods,
respectively. Finally, our exhaustive simulation results reveal the effect of
system parameters for various settings on the RBE-rate region of the SP mmWave
ISAC system and demonstrate a significantly enhanced performance compared to
the benchmarks.
