MinT - Best Known (84−81, 84, s)-Nets in Base 25

Best Known (84−81, 84, s)-Nets in Base 25

(84−81, 84, 52)-Net over F₂₅ — Constructive and digital

Digital (3, 84, 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]

(84−81, 84, 56)-Net over F₂₅ — Digital

Digital (3, 84, 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]

(84−81, 84, 102)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (3, 84, 103)-net over F₂₅, because

6 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]

(84−81, 84, 169)-Net in Base 25 — Upper bound on s

There is no (3, 84, 170)-net in base 25, because

1 times m-reduction [i] would yield (3, 83, 170)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁸³, 170, S₂₅, 80), but
  - the linear programming bound shows that MÂ â‰¥ 433348 260987 635051 497666 082664 253233 793870 559329 582169 538846 837609 651346 193409 689900 355589 149238 792756 602123 290576 855652 034282 684326 171875 / 3999 238211 111680 786259 > 25⁸³ [i]