MinT - Best Known (3, 82, s)-Nets in Base 25

Best Known (3, 82, s)-Nets in Base 25

(3, 82, 52)-Net over F₂₅ — Constructive and digital

Digital (3, 82, 52)-net over F₂₅, using

net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
  - a function field by GarcÃa and Quoos [i]

(3, 82, 56)-Net over F₂₅ — Digital

Digital (3, 82, 56)-net over F₂₅, using

net from sequence [i] based on digital (3, 55)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 56, using
  - fields by Serre with genus 3 or 4 [i]

(3, 82, 102)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (3, 82, 103)-net over F₂₅, because

4 times m-reduction [i] would yield digital (3, 78, 103)-net over F₂₅, but
- extracting embedded orthogonal array [i] would yield linear OA(25⁷⁸, 103, F₂₅, 75) (dual of [103, 25, 76]-code), but
  - residual code [i] would yield OA(25³, 27, S₂₅, 3), but
    - bound for OAs with index unity [i]

(3, 82, 170)-Net in Base 25 — Upper bound on s

There is no (3, 82, 171)-net in base 25, because

1 times m-reduction [i] would yield (3, 81, 171)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁸¹, 171, S₂₅, 78), but
  - the linear programming bound shows that MÂ â‰¥ 23205 784249 597023 772443 087743 632742 305389 821673 152514 637717 572494 013805 538409 460327 522440 800163 995577 864994 857009 151019 155979 156494 140625 / 133516 675673 119357 572553 > 25⁸¹ [i]