MinT - Best Known (108−77, 108, s)-Nets in Base 16

Best Known (108−77, 108, s)-Nets in Base 16

(108−77, 108, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 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]

(108−77, 108, 104)-Net in Base 16 — Constructive

(31, 108, 104)-net in base 16, using

2 times m-reduction [i] based on (31, 110, 104)-net in base 16, using
- base change [i] based on digital (9, 88, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(108−77, 108, 168)-Net over F₁₆ — Digital

Digital (31, 108, 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]

(108−77, 108, 2440)-Net in Base 16 — Upper bound on s

There is no (31, 108, 2441)-net in base 16, because

1 times m-reduction [i] would yield (31, 107, 2441)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 694 732469 465388 272244 140019 053306 510819 660273 271608 224903 812366 095678 041111 284795 365497 961047 233055 214698 461630 184764 756926 885296 > 16¹⁰⁷ [i]