MinT - Best Known (27, 27+79, s)-Nets in Base 16

Best Known (27, 27+79, s)-Nets in Base 16

(27, 27+79, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 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]

(27, 27+79, 76)-Net in Base 16 — Constructive

(27, 106, 76)-net in base 16, using

4 times m-reduction [i] based on (27, 110, 76)-net in base 16, using
- base change [i] based on digital (5, 88, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(27, 27+79, 156)-Net over F₁₆ — Digital

Digital (27, 106, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(27, 27+79, 1769)-Net in Base 16 — Upper bound on s

There is no (27, 106, 1770)-net in base 16, because

1 times m-reduction [i] would yield (27, 105, 1770)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 714969 736643 395162 057959 309653 657225 066115 101641 132530 244513 166967 349887 195795 802060 751030 270229 594556 744815 261990 792004 447076 > 16¹⁰⁵ [i]