MinT - Best Known (49, 106, s)-Nets in Base 27

Best Known (49, 106, s)-Nets in Base 27

(49, 106, 190)-Net over F₂₇ — Constructive and digital

Digital (49, 106, 190)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 38, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (11, 68, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(49, 106, 370)-Net in Base 27 — Constructive

(49, 106, 370)-net in base 27, using

t-expansion [i] based on (43, 106, 370)-net in base 27, using
- 2 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(49, 106, 421)-Net over F₂₇ — Digital

Digital (49, 106, 421)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 106, 101293)-Net in Base 27 — Upper bound on s

There is no (49, 106, 101294)-net in base 27, because

1 times m-reduction [i] would yield (49, 105, 101294)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964295 768412 979485 365059 598503 816557 925938 046028 191876 789309 061467 232901 216423 551534 671947 807673 993603 844715 647561 529868 916417 099408 208586 162900 372089 > 27¹⁰⁵ [i]