MinT - Best Known (37, 37+57, s)-Nets in Base 16

Best Known (37, 37+57, s)-Nets in Base 16

(37, 37+57, 103)-Net over F₁₆ — Constructive and digital

Digital (37, 94, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 31, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (6, 63, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(37, 37+57, 128)-Net in Base 16 — Constructive

(37, 94, 128)-net in base 16, using

2 times m-reduction [i] based on (37, 96, 128)-net in base 16, using
- base change [i] based on digital (5, 64, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(37, 37+57, 208)-Net over F₁₆ — Digital

Digital (37, 94, 208)-net over F₁₆, using

net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
  - Drinfeld modules of rank 1 [i]

(37, 37+57, 7506)-Net in Base 16 — Upper bound on s

There is no (37, 94, 7507)-net in base 16, because

1 times m-reduction [i] would yield (37, 93, 7507)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9645 563696 279626 734255 961411 211757 718436 271341 866580 971854 121043 097903 860633 042599 750515 693540 968444 189453 827216 > 16⁹³ [i]