MinT - Best Known (68−49, 68, s)-Nets in Base 27

Best Known (68−49, 68, s)-Nets in Base 27

(68−49, 68, 108)-Net over F₂₇ — Constructive and digital

Digital (19, 68, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 68, 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]

(68−49, 68, 116)-Net in Base 27 — Constructive

(19, 68, 116)-net in base 27, using

base change [i] based on digital (2, 51, 116)-net over F₈₁, using
- net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
    - a function field by SÃ©mirat [i]

(68−49, 68, 148)-Net over F₂₇ — Digital

Digital (19, 68, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 68, 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]

(68−49, 68, 3722)-Net in Base 27 — Upper bound on s

There is no (19, 68, 3723)-net in base 27, because

1 times m-reduction [i] would yield (19, 67, 3723)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 799436 009106 797281 492371 981698 540135 942730 631325 467985 382429 133583 617884 204750 866013 978873 106449 > 27⁶⁷ [i]