Optimality of Kløve Arrays within the Symmetric Kløve-Mossige Class

Abstract

Rajam$\unicode{228}$ki and Koivunen asked whether minimum-redundancy symmetric Kl$\unicode{248}$ve-Mossige arrays with contiguous sum co-arrays are always Kl$\unicode{248}$ve arrays. We give a computer-assisted proof that, at every fixed sensor count, every maximizing sensor set belongs to the Kl$\unicode{248}$ve class. The argument classifies the overlaps between a generator and its shifted reflection, then bounds the aperture of a hypothetical optimizer outside the Kl$\unicode{248}$ve class. Comparing these bounds with a classical Kl$\unicode{248}$ve construction settles all sensor counts at least 330. An exact integer certificate covers the remaining counts from 2 through 329 and matches every equality case to a Kl$\unicode{248}$ve array as a complete set. Consequently, optimization within the full symmetric Kl$\unicode{248}$ve-Mossige class reduces to the previously known search over Kl$\unicode{248}$ve parameters. The theorem concerns this specified class, rather than unrestricted sparse arrays or all restricted additive bases.

14 pages, 1 figure, 1 table. Verification code and certificates included as ancillary files. Code repository: https://github.com/Yan-ll9/kloeve-optimality