MinT - Best Known (33, 33+73, s)-Nets in Base 16

Best Known (33, 33+73, s)-Nets in Base 16

(33, 33+73, 65)-Net over F₁₆ — Constructive and digital

Digital (33, 106, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 106, 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]

(33, 33+73, 120)-Net in Base 16 — Constructive

(33, 106, 120)-net in base 16, using

4 times m-reduction [i] based on (33, 110, 120)-net in base 16, using
- base change [i] based on digital (11, 88, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(33, 33+73, 193)-Net over F₁₆ — Digital

Digital (33, 106, 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, 33+73, 3075)-Net in Base 16 — Upper bound on s

There is no (33, 106, 3076)-net in base 16, because

1 times m-reduction [i] would yield (33, 105, 3076)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 734756 415496 812858 055466 231935 800466 677041 844590 253978 911274 413928 160732 848980 753327 718175 903493 025691 593491 546975 746755 228416 > 16¹⁰⁵ [i]