MinT - Best Known (68−41, 68, s)-Nets in Base 5

Best Known (68−41, 68, s)-Nets in Base 5

(68−41, 68, 51)-Net over F₅ — Constructive and digital

Digital (27, 68, 51)-net over F₅, using

t-expansion [i] based on digital (22, 68, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(68−41, 68, 55)-Net over F₅ — Digital

Digital (27, 68, 55)-net over F₅, using

t-expansion [i] based on digital (23, 68, 55)-net over F₅, using
- net from sequence [i] based on digital (23, 54)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(68−41, 68, 441)-Net in Base 5 — Upper bound on s

There is no (27, 68, 442)-net in base 5, because

1 times m-reduction [i] would yield (27, 67, 442)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 69305 694458 993101 590602 815770 938060 875753 824065 > 5⁶⁷ [i]