MinT - Best Known (75−55, 75, s)-Nets in Base 27

Best Known (75−55, 75, s)-Nets in Base 27

(75−55, 75, 108)-Net over F₂₇ — Constructive and digital

Digital (20, 75, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 75, 108)-net over F₂₇, using
- net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
    - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(75−55, 75, 148)-Net over F₂₇ — Digital

Digital (20, 75, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 75, 148)-net over F₂₇, using
- net from sequence [i] based on digital (18, 147)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 148, using
    - a curve from the manYPoints database [i]

(75−55, 75, 3505)-Net in Base 27 — Upper bound on s

There is no (20, 75, 3506)-net in base 27, because

1 times m-reduction [i] would yield (20, 74, 3506)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 8395 289991 519201 090620 512464 738857 372132 745536 215236 507702 631767 010731 090231 989648 743571 972354 466425 670993 > 27⁷⁴ [i]