MinT - Best Known (39, 39+55, s)-Nets in Base 16

Best Known (39, 39+55, s)-Nets in Base 16

(39, 39+55, 130)-Net over F₁₆ — Constructive and digital

Digital (39, 94, 130)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 33, 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]
2. digital (6, 61, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(39, 39+55, 177)-Net in Base 16 — Constructive

(39, 94, 177)-net in base 16, using

2 times m-reduction [i] based on (39, 96, 177)-net in base 16, using
- base change [i] based on digital (7, 64, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(39, 39+55, 208)-Net over F₁₆ — Digital

Digital (39, 94, 208)-net over F₁₆, using

t-expansion [i] based on 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]

(39, 39+55, 10214)-Net in Base 16 — Upper bound on s

There is no (39, 94, 10215)-net in base 16, because

1 times m-reduction [i] would yield (39, 93, 10215)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9635 802104 830245 649519 919668 714062 672752 891917 625448 092333 513423 064721 425054 735701 112013 283326 158064 287281 686576 > 16⁹³ [i]