MinT - Best Known (62−35, 62, s)-Nets in Base 16

Best Known (62−35, 62, s)-Nets in Base 16

(62−35, 62, 110)-Net over F₁₆ — Constructive and digital

Digital (27, 62, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 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, 41, 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]

(62−35, 62, 129)-Net in Base 16 — Constructive

(27, 62, 129)-net in base 16, using

1 times m-reduction [i] based on (27, 63, 129)-net in base 16, using
- base change [i] based on digital (0, 36, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(62−35, 62, 156)-Net over F₁₆ — Digital

Digital (27, 62, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(62−35, 62, 161)-Net in Base 16

(27, 62, 161)-net in base 16, using

1 times m-reduction [i] based on (27, 63, 161)-net in base 16, using
- base change [i] based on digital (6, 42, 161)-net over F₆₄, using
  - net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(62−35, 62, 10002)-Net in Base 16 — Upper bound on s

There is no (27, 62, 10003)-net in base 16, because

1 times m-reduction [i] would yield (27, 61, 10003)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 28 277037 673180 147128 977464 639961 181174 846543 553898 658123 876348 885720 634366 > 16⁶¹ [i]