MinT - Best Known (108−75, 108, s)-Nets in Base 16

Best Known (108−75, 108, s)-Nets in Base 16

(108−75, 108, 65)-Net over F₁₆ — Constructive and digital

Digital (33, 108, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 108, 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]

(108−75, 108, 120)-Net in Base 16 — Constructive

(33, 108, 120)-net in base 16, using

2 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]

(108−75, 108, 193)-Net over F₁₆ — Digital

Digital (33, 108, 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]

(108−75, 108, 2944)-Net in Base 16 — Upper bound on s

There is no (33, 108, 2945)-net in base 16, because

1 times m-reduction [i] would yield (33, 107, 2945)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 698 058994 854747 058587 566311 573094 298180 817577 563637 825536 480721 692325 140976 090033 251620 444409 692723 877791 929903 135662 292969 742976 > 16¹⁰⁷ [i]