MinT - Best Known (33, 74, s)-Nets in Base 16

Best Known (33, 74, s)-Nets in Base 16

(33, 74, 130)-Net over F₁₆ — Constructive and digital

Digital (33, 74, 130)-net over F₁₆, using

1 times m-reduction [i] based on digital (33, 75, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 27, 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, 48, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(33, 74, 177)-Net in Base 16 — Constructive

(33, 74, 177)-net in base 16, using

4 times m-reduction [i] based on (33, 78, 177)-net in base 16, using
- base change [i] based on digital (7, 52, 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]

(33, 74, 193)-Net over F₁₆ — Digital

Digital (33, 74, 193)-net over F₁₆, using

net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
  - a curve from the manYPoints database [i]

(33, 74, 13737)-Net in Base 16 — Upper bound on s

There is no (33, 74, 13738)-net in base 16, because

1 times m-reduction [i] would yield (33, 73, 13738)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7961 889158 933016 172340 435381 046956 854097 869368 474359 672102 107521 768954 516947 849679 652276 > 16⁷³ [i]