MinT - Best Known (27, 27+45, s)-Nets in Base 27

Best Known (27, 27+45, s)-Nets in Base 27

(27, 27+45, 114)-Net over F₂₇ — Constructive and digital

Digital (27, 72, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 72, 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]

(27, 27+45, 172)-Net in Base 27 — Constructive

(27, 72, 172)-net in base 27, using

8 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]

(27, 27+45, 208)-Net over F₂₇ — Digital

Digital (27, 72, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 72, 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]

(27, 27+45, 244)-Net in Base 27

(27, 72, 244)-net in base 27, using

base change [i] based on digital (9, 54, 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]

(27, 27+45, 14485)-Net in Base 27 — Upper bound on s

There is no (27, 72, 14486)-net in base 27, because

1 times m-reduction [i] would yield (27, 71, 14486)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 423636 060009 845151 055377 647315 106226 310673 267699 354655 843600 984030 221471 087678 650196 683647 791991 601301 > 27⁷¹ [i]