  The design of communication signal sets is fundamentally a sphere packing
problem. It aims to identify a set of M points in an N -dimensional space, with
the objective of maximizing the separability of points that represent different
bits.In contrast, signals used for sensing targets should ideally be
asdeterministic as possible. This paper explores the inherent conflict and
trade-off between communication and sensing when these functions are combined
within the same signal set. We present a unified approach to signal design in
the time, frequency, and space domains for integrated sensing and communication
(ISAC), framing it as a modified sphere packing problem. Through adept formula
manipulation, this problem is transformed into a large-scale quadratic
constrained quadratic programming (QCQP) challenge. We propose an augmented
Lagrangian and dual ascent (ALDA) algorithm for iterative problem-solving. The
computational complexity of this approach is analyzed and found to be daunting
for large, high-dimensional signal set designs. To address this, we introduce a
bit-dimension-power splitting (BDPS) method. This method decomposes the
large-scale QCQP into a series of smaller-scale problems that can be solved
more efficiently and in parallel, significantly reducing the overall
computational load. Extensive simulations have been conducted to validate the
effectiveness of our proposed signal design methods in the context of ISAC.
