MinT - Best Known (19, 72, s)-Nets in Base 27

Best Known (19, 72, s)-Nets in Base 27

(19, 72, 108)-Net over F₂₇ — Constructive and digital

Digital (19, 72, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 72, 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]

(19, 72, 148)-Net over F₂₇ — Digital

Digital (19, 72, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 72, 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]

(19, 72, 3275)-Net in Base 27 — Upper bound on s

There is no (19, 72, 3276)-net in base 27, because

1 times m-reduction [i] would yield (19, 71, 3276)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 425620 286960 878215 824357 858644 934397 563556 063284 566976 653497 518833 123558 145908 649526 204172 842782 522601 > 27⁷¹ [i]