MinT - Best Known (38, 67, s)-Nets in Base 16

Best Known (38, 67, s)-Nets in Base 16

(38, 67, 522)-Net over F₁₆ — Constructive and digital

Digital (38, 67, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (38, 68, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 34, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(38, 67, 644)-Net over F₁₆ — Digital

Digital (38, 67, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁶⁷, 644, F₁₆, 2, 29) (dual of [(644, 2), 1221, 30]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁶⁶, 644, F₁₆, 2, 29) (dual of [(644, 2), 1222, 30]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁶², 642, F₁₆, 2, 29) (dual of [(642, 2), 1222, 30]-NRT-code), using
    - extracting embedded OOA [i] based on digital (33, 62, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 31, 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, 67, 191388)-Net in Base 16 — Upper bound on s

There is no (38, 67, 191389)-net in base 16, because

1 times m-reduction [i] would yield (38, 66, 191389)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 644425 523800 801728 753812 874440 171614 184731 329555 584691 559710 238489 233249 557816 > 16⁶⁶ [i]