MinT - Best Known (32, 89, s)-Nets in Base 27

Best Known (32, 89, s)-Nets in Base 27

Digital (32, 89, 114)-net over F₂₇, using

(32, 89, 172)-net in base 27, using

11 times m-reduction [i] based on (32, 100, 172)-net in base 27, using
- base change [i] based on digital (7, 75, 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]

Digital (32, 89, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 89, 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]

(32, 89, 244)-net in base 27, using

3 times m-reduction [i] based on (32, 92, 244)-net in base 27, using
- base change [i] based on digital (9, 69, 244)-net over F₈₁, using
  - net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

There is no (32, 89, 13682)-net in base 27, because

1 times m-reduction [i] would yield (32, 88, 13682)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 912529 628780 572455 079491 016787 150991 357107 557654 778760 385967 446371 412177 296697 192451 954447 959506 284423 899166 237146 353177 103385 > 27⁸⁸ [i]