MinT - Best Known (27, 27+37, s)-Nets in Base 5

Best Known (27, 27+37, s)-Nets in Base 5

(27, 27+37, 51)-Net over F₅ — Constructive and digital

Digital (27, 64, 51)-net over F₅, using

t-expansion [i] based on digital (22, 64, 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]

(27, 27+37, 55)-Net over F₅ — Digital

Digital (27, 64, 55)-net over F₅, using

t-expansion [i] based on digital (23, 64, 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]

(27, 27+37, 514)-Net in Base 5 — Upper bound on s

There is no (27, 64, 515)-net in base 5, because

1 times m-reduction [i] would yield (27, 63, 515)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 108 755363 971792 074493 288879 824390 920220 794425 > 5⁶³ [i]