MinT - Best Known (78−55, 78, s)-Nets in Base 27

Best Known (78−55, 78, s)-Nets in Base 27

(78−55, 78, 114)-Net over F₂₇ — Constructive and digital

Digital (23, 78, 114)-net over F₂₇, using

net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
  - an explicitly constructive algebraic function field [i]

(78−55, 78, 116)-Net in Base 27 — Constructive

(23, 78, 116)-net in base 27, using

6 times m-reduction [i] based on (23, 84, 116)-net in base 27, using
- base change [i] based on digital (2, 63, 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]

(78−55, 78, 163)-Net over F₂₇ — Digital

Digital (23, 78, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 78, 163)-net over F₂₇, using
- net from sequence [i] based on digital (21, 162)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 21 and N(F) ≥ 163, using
    - a curve from the manYPoints database [i]

(78−55, 78, 5061)-Net in Base 27 — Upper bound on s

There is no (23, 78, 5062)-net in base 27, because

1 times m-reduction [i] would yield (23, 77, 5062)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 164 607584 570791 593644 741889 461881 415334 841841 034056 576392 136330 365707 073934 348149 815997 985268 014547 958127 180417 > 27⁷⁷ [i]