MinT - Best Known (80−43, 80, s)-Nets in Base 16

Best Known (80−43, 80, s)-Nets in Base 16

(80−43, 80, 130)-Net over F₁₆ — Constructive and digital

Digital (37, 80, 130)-net over F₁₆, using

7 times m-reduction [i] based on digital (37, 87, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 31, 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, 56, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(80−43, 80, 177)-Net in Base 16 — Constructive

(37, 80, 177)-net in base 16, using

10 times m-reduction [i] based on (37, 90, 177)-net in base 16, using
- base change [i] based on digital (7, 60, 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]

(80−43, 80, 209)-Net over F₁₆ — Digital

Digital (37, 80, 209)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(80−43, 80, 225)-Net in Base 16

(37, 80, 225)-net in base 16, using

1 times m-reduction [i] based on (37, 81, 225)-net in base 16, using
- base change [i] based on digital (10, 54, 225)-net over F₆₄, using
  - net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
      - a curve from the manYPoints database [i]

(80−43, 80, 19585)-Net in Base 16 — Upper bound on s

There is no (37, 80, 19586)-net in base 16, because

1 times m-reduction [i] would yield (37, 79, 19586)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133579 353307 487902 863418 010966 166534 974855 088488 528050 746552 095432 110918 840337 194579 883865 661216 > 16⁷⁹ [i]