MinT - Best Known (78−51, 78, s)-Nets in Base 27

Best Known (78−51, 78, s)-Nets in Base 27

(78−51, 78, 114)-Net over F₂₇ — Constructive and digital

Digital (27, 78, 114)-net over F₂₇, using

t-expansion [i] based on 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]

(78−51, 78, 172)-Net in Base 27 — Constructive

(27, 78, 172)-net in base 27, using

2 times m-reduction [i] based on (27, 80, 172)-net in base 27, using
- base change [i] based on digital (7, 60, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(78−51, 78, 208)-Net over F₂₇ — Digital

Digital (27, 78, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 78, 208)-net over F₂₇, using
- net from sequence [i] based on digital (24, 207)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 24 and N(F) ≥ 208, using
    - an algebraic function field by Beelen and Pellikaan [i]

(78−51, 78, 10015)-Net in Base 27 — Upper bound on s

There is no (27, 78, 10016)-net in base 27, because

1 times m-reduction [i] would yield (27, 77, 10016)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 164 090359 238084 410211 614207 247970 896988 282616 251261 917645 188340 604273 342943 425201 710419 979235 686306 968632 813377 > 27⁷⁷ [i]