Reflection of optimal supports for k-wise independent bits
Abstract
For even k, we derive a reflection identity for the basic count distributions in the problem of maximizing the probability that all n k-wise independent Bernoulli variables equal one. After separating the distinguished node n, the other support nodes reflect by s -> n-1-s while the Bernoulli parameter changes from p to 1-p. Corresponding basic weights differ by an explicit positive factor. Thus interior feasibility sets, and local changes of optimal support with their multiplicities, are reflected. This gives a proof of the exceptional-point reflection in Conjecture 5.11 of Berend, Ernst, Kontorovich and Kumar. The argument uses Lagrange interpolation and an elementary binomial reweighting identity; it does not require a conjectured ordering or connectedness of the feasibility sets.
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BibTeX
@misc{hermann2026reflection,
title = {{Reflection of optimal supports for k-wise independent bits}},
author = {Roy Hermann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25595},
eprint = {2609.25595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25595},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reflection of optimal supports for k-wise independent bits
AU - Roy Hermann
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25595
UR - https://arxiv.org/abs/2609.25595
AB - For even k, we derive a reflection identity for the basic count distributions in the problem of maximizing the probability that all n k-wise independent Bernoulli variables equal one. After separating the distinguished node n, the other support nodes reflect by s -> n-1-s while the Bernoulli parameter changes from p to 1-p. Corresponding basic weights differ by an explicit positive factor. Thus interior feasibility sets, and local changes of optimal support with their multiplicities, are reflected. This gives a proof of the exceptional-point reflection in Conjecture 5.11 of Berend, Ernst, Kontorovich and Kumar. The argument uses Lagrange interpolation and an elementary binomial reweighting identity; it does not require a conjectured ordering or connectedness of the feasibility sets.
ER -