MinT - Best Known (56−27, 56, s)-Nets in Base 16

Best Known (56−27, 56, s)-Nets in Base 16

(56−27, 56, 516)-Net over F₁₆ — Constructive and digital

Digital (29, 56, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 28, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(56−27, 56, 578)-Net over F₁₆ — Digital

Digital (29, 56, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 28, 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]

(56−27, 56, 46946)-Net in Base 16 — Upper bound on s

There is no (29, 56, 46947)-net in base 16, because

1 times m-reduction [i] would yield (29, 55, 46947)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 685131 240543 950052 217316 511726 276316 265441 344145 853883 347577 860816 > 16⁵⁵ [i]