An exact compression-and-lifting procedure and D-optimal designs of orders 198 and 170
Abstract
We develop a compression-and-lifting procedure for constructing D-optimal complementary pairs on finite abelian groups of odd order. The procedure separates the Fourier conditions preserved by compression from those needed to recover the original binary sequences. For groups with a cyclic factor of order nine, integer matching conditions separate compatible classes of local sign assignments. Their exact sizes can be computed before full sequences are generated, allowing searches to discard incompatible pairs and select manageable classes. A bound on cosets on which a sequence is constant also excludes certain automorphisms as fixing symmetries. Applying the method yields supplementary difference sets with parameters $(99;43,42;36)$ and $(85;39,34;31)$, resolving two previously undecided parameter sets. These give the first D-optimal matrix of order $198$ and constructions with new parameters at the known order $170$. For the successful length-$99$ compression, the exact class condition removes 99.9888% of candidate pairs while preserving every D-optimal lift. A reflection-invariant construction at length $85$ gives seven supplementary difference set classes and fourteen matrix classes under signed row and column permutations, paired by transposition. The class reductions guide computation; finding suitable compressions and matching their lifts remain search problems.
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BibTeX
@misc{georgiou2026exact,
title = {{An exact compression-and-lifting procedure and D-optimal designs of orders 198 and 170}},
author = {Stelios D. Georgiou and Stella Stylianou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33418},
eprint = {2610.33418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33418},
}
RIS
TY - UNPB
TI - An exact compression-and-lifting procedure and D-optimal designs of orders 198 and 170
AU - Stelios D. Georgiou
AU - Stella Stylianou
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33418
UR - https://arxiv.org/abs/2610.33418
AB - We develop a compression-and-lifting procedure for constructing D-optimal complementary pairs on finite abelian groups of odd order. The procedure separates the Fourier conditions preserved by compression from those needed to recover the original binary sequences. For groups with a cyclic factor of order nine, integer matching conditions separate compatible classes of local sign assignments. Their exact sizes can be computed before full sequences are generated, allowing searches to discard incompatible pairs and select manageable classes. A bound on cosets on which a sequence is constant also excludes certain automorphisms as fixing symmetries. Applying the method yields supplementary difference sets with parameters $(99;43,42;36)$ and $(85;39,34;31)$, resolving two previously undecided parameter sets. These give the first D-optimal matrix of order $198$ and constructions with new parameters at the known order $170$. For the successful length-$99$ compression, the exact class condition removes 99.9888% of candidate pairs while preserving every D-optimal lift. A reflection-invariant construction at length $85$ gives seven supplementary difference set classes and fourteen matrix classes under signed row and column permutations, paired by transposition. The class reductions guide computation; finding suitable compressions and matching their lifts remain search problems.
ER -