MinT - Best Known (68−39, 68, s)-Nets in Base 16

Best Known (68−39, 68, s)-Nets in Base 16

(68−39, 68, 110)-Net over F₁₆ — Constructive and digital

Digital (29, 68, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 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, 45, 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]

(68−39, 68, 128)-Net in Base 16 — Constructive

(29, 68, 128)-net in base 16, using

4 times m-reduction [i] based on (29, 72, 128)-net in base 16, using
- base change [i] based on digital (5, 48, 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]

(68−39, 68, 161)-Net over F₁₆ — Digital

Digital (29, 68, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(68−39, 68, 9305)-Net in Base 16 — Upper bound on s

There is no (29, 68, 9306)-net in base 16, because

1 times m-reduction [i] would yield (29, 67, 9306)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 474 496520 415600 283619 577411 186968 718497 190761 698522 742303 140856 078899 152430 962436 > 16⁶⁷ [i]