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In many applications including communications, one may encounter a linearmodel where the parameter vector $\hbx$ is an integer vector in a box. Toestimate $\hbx$, a typical method is to solve a box-constrained integer leastsquares (BILS) problem. However, due to its high complexity, thebox-constrained Babai integer point $\x^\sBB$ is commonly used as a suboptimalsolution. In this paper, we first derive formulas for the success probability$P^\sBB$ of $\x^\sBB$ and the success probability $P^\sOB$ of the ordinaryBabai integer point $\x^\sOB$ when $\hbx$ is uniformly distributed over theconstraint box. Some properties of $P^\sBB$ and $P^\sOB$ and the relationshipbetween them are studied. Then, we investigate the effects of some columnpermutation strategies on $\P^\sBB$. In addition to V-BLAST and SQRD, we alsoconsider the permutation strategy involved in the LLL lattice reduction, to bereferred to as LLL-P. On the one hand, we show that when the noise isrelatively small, LLL-P always increases $P^\sBB$ and argue why both V-BLASTand SQRD often increase $P^\sBB$; and on the other hand, we show that when thenoise is relatively large, LLL-P always decreases $P^\sBB$ and argue why bothV-BLAST and SQRD often decrease $P^\sBB$. We also derive a column permutationinvariant bound on $P^\sBB$, which is an upper bound and a lower bound underthese two opposite conditions, respectively. Numerical results demonstrate ourfindings. Finally, we consider a conjecture concerning $\x^\sOB$ proposed by Maet al. We first construct an example to show that the conjecture does not holdin general, and then show that it does hold under some conditions.

Success probability of the Babai estimators for box-constrained integer linear models