An efficient construction strategy for minimum redundancy linear arrays based on linear array formulations

The acquisition of large‑element minimum redundancy linear arrays (MRLAs) remains a challenging problem in radar, microwave radiometer, and wireless communication systems. Existing computational approaches often require prohibitively long runtimes, yields only limited number of solutions and may fail to identify application-specific optimal array configurations. To address this issue, this study analyzes the defining conditions of four mathematical constructs: (i) restricted difference bases with the minimum number of elements for a given baseline length <italic>L</italic>; (ii) MRLAs with maximum contiguous baseline length <italic>L</italic>; (iii) scale values of perfect sparse rulers of length <italic>L</italic>, and (iv) vertex labelings of minimal graceful graphs with <italic>L</italic> edges. Through cyclic reasoning, their mathematical equivalence among these four formulations is rigorously established. Several theoretical properties are further derived. First, linear arrays exist in symmetric pairs. Second, the redundancy <italic>R</italic> of a linear array satisfies <italic>R</italic> ≥ 1, with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>R</mml:mi><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math></inline-formula> when the number of elements exceeds four. Third, if an MRLA with maximum contiguous baseline length <italic>L</italic> contains <italic>n</italic> elements, an MRLA with maximum baseline length <italic>L</italic> + 1 contains no more than <italic>n</italic> + 1 elements. Based on large‑scale MRLA data analysis, a practical hypothesis is proposed: linear arrays with redundancy <italic>R</italic> ≤ 1.5 may be regarded as effectively minimum‑redundancy linear arrays. In addition, two novel types of analytical formulas for linear arrays are introduced, enabling efficient screening of infinitely MRLA configuration patterns—each corresponding to scale values of perfect sparse rulers—and allowing flexible adjustment of redundancy thresholds according to application requirements. These results provide theoretical support for both the practical deployment of MRLAs and the systematic design of perfect sparse rulers.