MinT - Best Known (37, 37+55, s)-Nets in Base 16

Best Known (37, 37+55, s)-Nets in Base 16

(37, 37+55, 110)-Net over F₁₆ — Constructive and digital

Digital (37, 92, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 31, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 61, 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+55, 128)-Net in Base 16 — Constructive

(37, 92, 128)-net in base 16, using

4 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+55, 208)-Net over F₁₆ — Digital

Digital (37, 92, 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+55, 8315)-Net in Base 16 — Upper bound on s

There is no (37, 92, 8316)-net in base 16, because

1 times m-reduction [i] would yield (37, 91, 8316)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 37 671506 814489 713041 509704 853188 599391 579281 925142 910098 823615 777422 979393 568522 152596 856434 028191 722893 966081 > 16⁹¹ [i]