MinT - Best Known (39, 39+61, s)-Nets in Base 16

Best Known (39, 39+61, s)-Nets in Base 16

(39, 39+61, 103)-Net over F₁₆ — Constructive and digital

Digital (39, 100, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 33, 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, 67, 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]

(39, 39+61, 128)-Net in Base 16 — Constructive

(39, 100, 128)-net in base 16, using

2 times m-reduction [i] based on (39, 102, 128)-net in base 16, using
- base change [i] based on digital (5, 68, 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]

(39, 39+61, 208)-Net over F₁₆ — Digital

Digital (39, 100, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 100, 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]

(39, 39+61, 7539)-Net in Base 16 — Upper bound on s

There is no (39, 100, 7540)-net in base 16, because

1 times m-reduction [i] would yield (39, 99, 7540)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 161689 829092 086612 106971 652990 595463 354897 710384 060005 835859 897115 974621 435002 509965 861505 079031 178835 497273 555766 472376 > 16⁹⁹ [i]