MinT - Best Known (90−65, 90, s)-Nets in Base 27

Best Known (90−65, 90, s)-Nets in Base 27

(90−65, 90, 114)-Net over F₂₇ — Constructive and digital

Digital (25, 90, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 90, 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]

(90−65, 90, 116)-Net in Base 27 — Constructive

(25, 90, 116)-net in base 27, using

2 times m-reduction [i] based on (25, 92, 116)-net in base 27, using
- base change [i] based on digital (2, 69, 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]

(90−65, 90, 208)-Net over F₂₇ — Digital

Digital (25, 90, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 90, 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]

(90−65, 90, 4691)-Net in Base 27 — Upper bound on s

There is no (25, 90, 4692)-net in base 27, because

1 times m-reduction [i] would yield (25, 89, 4692)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 652432 966009 579398 484499 523092 720578 143619 976436 046441 075547 033938 424932 100938 144623 803580 490064 090412 061082 029161 616187 358465 > 27⁸⁹ [i]