MinT - Best Known (84−41, 84, s)-Nets in Base 16

Best Known (84−41, 84, s)-Nets in Base 16

(84−41, 84, 516)-Net over F₁₆ — Constructive and digital

Digital (43, 84, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 42, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(84−41, 84, 578)-Net over F₁₆ — Digital

Digital (43, 84, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 42, 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]

(84−41, 84, 54982)-Net in Base 16 — Upper bound on s

There is no (43, 84, 54983)-net in base 16, because

1 times m-reduction [i] would yield (43, 83, 54983)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8749 441676 452133 980222 902331 349388 840603 583318 011521 656507 133907 763493 007472 374260 255400 198118 693276 > 16⁸³ [i]