MinT - Best Known (108−81, 108, s)-Nets in Base 16

Best Known (108−81, 108, s)-Nets in Base 16

(108−81, 108, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 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]

(108−81, 108, 76)-Net in Base 16 — Constructive

(27, 108, 76)-net in base 16, using

2 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]

(108−81, 108, 156)-Net over F₁₆ — Digital

Digital (27, 108, 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]

(108−81, 108, 1726)-Net in Base 16 — Upper bound on s

There is no (27, 108, 1727)-net in base 16, because

1 times m-reduction [i] would yield (27, 107, 1727)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 699 137014 197023 434151 533525 915877 125697 286427 198354 703520 271643 021835 099878 361912 650233 069399 632417 223960 366778 905317 782970 298951 > 16¹⁰⁷ [i]