MinT - Best Known (128−61, 128, s)-Nets in Base 16

Best Known (128−61, 128, s)-Nets in Base 16

(128−61, 128, 520)-Net over F₁₆ — Constructive and digital

Digital (67, 128, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 64, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(128−61, 128, 642)-Net over F₁₆ — Digital

Digital (67, 128, 642)-net over F₁₆, using

2 times m-reduction [i] based on digital (67, 130, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 65, 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]

(128−61, 128, 100477)-Net in Base 16 — Upper bound on s

There is no (67, 128, 100478)-net in base 16, because

1 times m-reduction [i] would yield (67, 127, 100478)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 838 197615 285993 803924 823339 041519 151772 208694 737363 629846 707539 123801 625027 188791 664971 637779 711451 943765 652886 010221 083497 983911 500095 259592 192385 592976 > 16¹²⁷ [i]