MinT - Best Known (65−29, 65, s)-Nets in Base 5

Best Known (65−29, 65, s)-Nets in Base 5

(65−29, 65, 104)-Net over F₅ — Constructive and digital

Digital (36, 65, 104)-net over F₅, using

1 times m-reduction [i] based on digital (36, 66, 104)-net over F₅, using
- trace code for nets [i] based on digital (3, 33, 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]

(65−29, 65, 112)-Net over F₅ — Digital

Digital (36, 65, 112)-net over F₅, using

1 times m-reduction [i] based on digital (36, 66, 112)-net over F₅, using
- trace code for nets [i] based on digital (3, 33, 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]

(65−29, 65, 2359)-Net in Base 5 — Upper bound on s

There is no (36, 65, 2360)-net in base 5, because

1 times m-reduction [i] would yield (36, 64, 2360)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 543 086328 521663 726314 951120 342449 553726 861185 > 5⁶⁴ [i]