MinT - Best Known (37, 99, s)-Nets in Base 16

Best Known (37, 99, s)-Nets in Base 16

(37, 99, 82)-Net over F₁₆ — Constructive and digital

Digital (37, 99, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 31, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 68, 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, 99, 120)-Net in Base 16 — Constructive

(37, 99, 120)-net in base 16, using

31 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
- base change [i] based on digital (11, 104, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(37, 99, 208)-Net over F₁₆ — Digital

Digital (37, 99, 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, 99, 5781)-Net in Base 16 — Upper bound on s

There is no (37, 99, 5782)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 161390 951059 542353 761387 442092 114418 652813 691443 962213 889930 439040 521899 132002 080076 427211 002467 926069 799092 617863 016256 > 16⁹⁹ [i]