Let ε > 0 be a fixed real number, ⊂ R s be a full rank lattice with determinant Δ ∈ Q . We call this lattice ε -regular if λ 1 () > Δ 1/ s ( h (Δ)) – ε , where λ 1 () is the length of the shortest nonzero vector of and h (Δ) is the maximum of absolute values of the numerator and the denominator of the irreducible rational fraction for Δ. In this paper, we consider two full rank lattices in the space R s : the lattice ℒ( a , W ) connected with the linear congruent sequence ( x N ), x N +1 = ax N (mod W ), N =1 ,2, . . . , (1) and the lattice ℒ ∗ ( a , W ) dual to ℒ( a , W ). There is a conjecture which states that for any natural number s , any real number 0 < ε < ε 0 ( s ), and any natural number W > W 0 ( s , ε ), the lattices ℒ( a , W ) and ℒ ∗ ( a , W ) are ε -regular for all a = 0 ,1, . . . , W – 1 excluding some set of numbers a of cardinality at most W 1– ε . In the case s = 3, A. M. Frieze, J. Hestad, R. Kannan, J. C. Lagarias, and A. Shamir in a paper published in 1988 proved a more weak assertion (in their estimate the number of exceptional values a is at most W 1– ε /2 ). Using the methods of this paper, it is not difficult to prove the conjecture for s = 1 and s = 2. In our paper, we prove the conjecture for s = 4. With the help of our methods we improve the result of the paper mentioned above and prove the conjecture for s = 3. Our result can be applied to the reconstruction of a linear congruent sequence (1) if the high-order bits of its first s elements are given.