MinT - Best Known (33, 33+69, s)-Nets in Base 27

Best Known (33, 33+69, s)-Nets in Base 27

(33, 33+69, 114)-Net over F₂₇ — Constructive and digital

Digital (33, 102, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 102, 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]

(33, 33+69, 172)-Net in Base 27 — Constructive

(33, 102, 172)-net in base 27, using

2 times m-reduction [i] based on (33, 104, 172)-net in base 27, using
- base change [i] based on digital (7, 78, 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]

(33, 33+69, 220)-Net over F₂₇ — Digital

Digital (33, 102, 220)-net over F₂₇, using

net from sequence [i] based on digital (33, 219)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 33 and N(F) ≥ 220, using
  - Drinfeld modules of rank 1 [i]

(33, 33+69, 9282)-Net in Base 27 — Upper bound on s

There is no (33, 102, 9283)-net in base 27, because

1 times m-reduction [i] would yield (33, 101, 9283)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 703563 500957 403375 195882 338300 761580 329190 033114 296042 543575 568740 563624 607583 214768 131945 995144 665643 432287 481695 864168 246233 188443 725314 514101 > 27¹⁰¹ [i]