MinT - Best Known (36, 36+33, s)-Nets in Base 16

Best Known (36, 36+33, s)-Nets in Base 16

(36, 36+33, 516)-Net over F₁₆ — Constructive and digital

Digital (36, 69, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (36, 70, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 35, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(36, 36+33, 578)-Net over F₁₆ — Digital

Digital (36, 69, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (36, 70, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 35, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(36, 36+33, 59414)-Net in Base 16 — Upper bound on s

There is no (36, 69, 59415)-net in base 16, because

1 times m-reduction [i] would yield (36, 68, 59415)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7588 919649 477098 710504 708655 132727 763423 758926 157474 425164 633228 934806 845692 762101 > 16⁶⁸ [i]