MinT - Best Known (85−61, 85, s)-Nets in Base 5

Best Known (85−61, 85, s)-Nets in Base 5

(85−61, 85, 51)-Net over F₅ — Constructive and digital

Digital (24, 85, 51)-net over F₅, using

t-expansion [i] based on digital (22, 85, 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]

(85−61, 85, 55)-Net over F₅ — Digital

Digital (24, 85, 55)-net over F₅, using

t-expansion [i] based on digital (23, 85, 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]

(85−61, 85, 251)-Net in Base 5 — Upper bound on s

There is no (24, 85, 252)-net in base 5, because

1 times m-reduction [i] would yield (24, 84, 252)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 55156 156800 002435 993164 474268 018401 828954 930496 517659 189825 > 5⁸⁴ [i]