MinT - Best Known (31, 31+85, s)-Nets in Base 16

Best Known (31, 31+85, s)-Nets in Base 16

(31, 31+85, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 116, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 116, 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, 31+85, 98)-Net in Base 16 — Constructive

(31, 116, 98)-net in base 16, using

4 times m-reduction [i] based on (31, 120, 98)-net in base 16, using
- base change [i] based on digital (7, 96, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(31, 31+85, 168)-Net over F₁₆ — Digital

Digital (31, 116, 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, 31+85, 2158)-Net in Base 16 — Upper bound on s

There is no (31, 116, 2159)-net in base 16, because

1 times m-reduction [i] would yield (31, 115, 2159)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 034222 421373 516925 912276 470552 053591 344059 803614 272299 620890 133611 391723 109377 495019 866354 922282 283530 902572 517522 880620 348711 465426 829671 > 16¹¹⁵ [i]