MinT - Best Known (33, 106, s)-Nets in Base 27

Best Known (33, 106, s)-Nets in Base 27

(33, 106, 114)-Net over F₂₇ — Constructive and digital

Digital (33, 106, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 106, 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, 106, 160)-Net in Base 27 — Constructive

(33, 106, 160)-net in base 27, using

t-expansion [i] based on (32, 106, 160)-net in base 27, using
- 2 times m-reduction [i] based on (32, 108, 160)-net in base 27, using
  - base change [i] based on digital (5, 81, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(33, 106, 220)-Net over F₂₇ — Digital

Digital (33, 106, 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, 106, 8195)-Net in Base 27 — Upper bound on s

There is no (33, 106, 8196)-net in base 27, because

1 times m-reduction [i] would yield (33, 105, 8196)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 968520 295382 563623 156774 140184 124883 715777 460042 547481 566568 577801 918204 981908 409811 473029 659864 353349 975256 237826 113338 715567 410097 806002 800544 430273 > 27¹⁰⁵ [i]