MinT - Best Known (15, 15+31, s)-Nets in Base 27

Best Known (15, 15+31, s)-Nets in Base 27

(15, 15+31, 96)-Net over F₂₇ — Constructive and digital

Digital (15, 46, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 46, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(15, 15+31, 116)-Net in Base 27 — Constructive

(15, 46, 116)-net in base 27, using

6 times m-reduction [i] based on (15, 52, 116)-net in base 27, using
- base change [i] based on digital (2, 39, 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]

(15, 15+31, 136)-Net over F₂₇ — Digital

Digital (15, 46, 136)-net over F₂₇, using

t-expansion [i] based on digital (13, 46, 136)-net over F₂₇, using
- net from sequence [i] based on digital (13, 135)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 13 and N(F) ≥ 136, using
    - a curve from the manYPoints database [i]

(15, 15+31, 4855)-Net in Base 27 — Upper bound on s

There is no (15, 46, 4856)-net in base 27, because

1 times m-reduction [i] would yield (15, 45, 4856)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 25848 403301 971541 108768 738512 323432 199389 484041 383856 447154 283745 > 27⁴⁵ [i]