MinT - Best Known (62−39, 62, s)-Nets in Base 16

Best Known (62−39, 62, s)-Nets in Base 16

(62−39, 62, 66)-Net over F₁₆ — Constructive and digital

Digital (23, 62, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (2, 41, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(62−39, 62, 104)-Net in Base 16 — Constructive

(23, 62, 104)-net in base 16, using

8 times m-reduction [i] based on (23, 70, 104)-net in base 16, using
- base change [i] based on digital (9, 56, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(62−39, 62, 129)-Net over F₁₆ — Digital

Digital (23, 62, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 62, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(62−39, 62, 3871)-Net in Base 16 — Upper bound on s

There is no (23, 62, 3872)-net in base 16, because

1 times m-reduction [i] would yield (23, 61, 3872)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 28 395292 097909 554657 040709 041329 728389 463627 093035 649152 653473 580835 073571 > 16⁶¹ [i]