Prove lcrm equality for the proposed non-diagonal constructions across general exponents and dimensions

Prove that for the non-diagonal matrix constructions (obtained by following Guo and Xia 2025) associated with a modulus determinant qi having prime factorization qi = p1^{n1} p2^{n2} ... pk^{nk} in D dimensions, the least common right multiple of the D matrices in the group equals the diagonal matrix diag(qi, ..., qi) for arbitrary exponent patterns {n_j} and any dimension D (beyond those verified by simulation).

Background

In Section 3.3 the authors discuss replacing diagonal sampling matrices with non-diagonal ones that are more balanced per dimension while preserving determinant and pairwise coprimality properties. For the special case where all exponents equal D, they show how to construct D non-diagonal, pairwise co-prime matrices whose lcrm is diag(qi, ..., qi).

For more general exponent patterns {n_j}, they explain that similar non-diagonal constructions can be obtained and are pairwise co-prime, and simulations up to D ≤ 75 indicate that the lcrm still equals diag(qi, ..., qi). However, a complete theoretical proof covering all exponent patterns and dimensions is not yet established.

References

For other cases of the exponents n_j, similar non-diagonal constructions can still be obtained by following the same approach in . They also lead to more balanced per-dimension sampling rates compared with the diagonal ones. We can also show that the resulting matrices are pairwise co-prime. Although our simulations indicate that their lcrm still equals \mathrm{diag}(q_i,\ldots,q_i) for dimension D\leq 75, a complete theoretical proof remains open and will be left for future work.

— Robust Multidimensional Chinese Remainder Theorem (MD-CRT) with Non-Diagonal Moduli and Multi-Stage Framework  (2604.00995 - Guo et al., 1 Apr 2026) in Section 3.3, Sampling Advantage of Non-Diagonal Matrix Moduli (end of subsection)