MinT - Best Known (31, 31+29, s)-Nets in Base 16

Best Known (31, 31+29, s)-Nets in Base 16

(31, 31+29, 516)-Net over F₁₆ — Constructive and digital

Digital (31, 60, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(31, 31+29, 578)-Net over F₁₆ — Digital

Digital (31, 60, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 30, 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]

(31, 31+29, 47841)-Net in Base 16 — Upper bound on s

There is no (31, 60, 47842)-net in base 16, because

1 times m-reduction [i] would yield (31, 59, 47842)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 110444 583273 148163 491591 144382 351395 767733 459143 675542 390075 689131 108496 > 16⁵⁹ [i]