MinT - Best Known (27, 27+79, s)-Nets in Base 5

Best Known (27, 27+79, s)-Nets in Base 5

(27, 27+79, 51)-Net over F₅ — Constructive and digital

Digital (27, 106, 51)-net over F₅, using

t-expansion [i] based on digital (22, 106, 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+79, 55)-Net over F₅ — Digital

Digital (27, 106, 55)-net over F₅, using

t-expansion [i] based on digital (23, 106, 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+79, 265)-Net in Base 5 — Upper bound on s

There is no (27, 106, 266)-net in base 5, because

1 times m-reduction [i] would yield (27, 105, 266)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 26 564485 070440 511733 847812 160888 031899 701230 561466 254627 484176 421489 681225 > 5¹⁰⁵ [i]