MinT - Best Known (31, 88, s)-Nets in Base 16

Best Known (31, 88, s)-Nets in Base 16

(31, 88, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 88, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 88, 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]

(31, 88, 120)-Net in Base 16 — Constructive

(31, 88, 120)-net in base 16, using

12 times m-reduction [i] based on (31, 100, 120)-net in base 16, using
- base change [i] based on digital (11, 80, 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]

(31, 88, 168)-Net over F₁₆ — Digital

Digital (31, 88, 168)-net over F₁₆, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(31, 88, 4136)-Net in Base 16 — Upper bound on s

There is no (31, 88, 4137)-net in base 16, because

1 times m-reduction [i] would yield (31, 87, 4137)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 573 656605 664075 146044 118065 055022 472511 519972 022198 743874 302061 153193 553466 774275 187893 294362 450253 171816 > 16⁸⁷ [i]