MinT - Best Known (38−16, 38, s)-Nets in Base 16

Best Known (38−16, 38, s)-Nets in Base 16

(38−16, 38, 520)-Net over F₁₆ — Constructive and digital

Digital (22, 38, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 19, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(38−16, 38, 643)-Net over F₁₆ — Digital

Digital (22, 38, 643)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16³⁸, 643, F₁₆, 2, 16) (dual of [(643, 2), 1248, 17]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(16³⁶, 642, F₁₆, 2, 16) (dual of [(642, 2), 1248, 17]-NRT-code), using
  - extracting embedded OOA [i] based on digital (20, 36, 642)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 18, 321)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
        sharp bound on the number of rational points on curves with genus 2 [i]

(38−16, 38, 131569)-Net in Base 16 — Upper bound on s

There is no (22, 38, 131570)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5709 144717 496493 697660 966285 760651 320809 432026 > 16³⁸ [i]