MinT - Best Known (45, 45+61, s)-Nets in Base 27

Best Known (45, 45+61, s)-Nets in Base 27

(45, 45+61, 166)-Net over F₂₇ — Constructive and digital

Digital (45, 106, 166)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (8, 69, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(45, 45+61, 288)-Net over F₂₇ — Digital

Digital (45, 106, 288)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+61, 370)-Net in Base 27 — Constructive

(45, 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]

(45, 45+61, 47363)-Net in Base 27 — Upper bound on s

There is no (45, 106, 47364)-net in base 27, because

1 times m-reduction [i] would yield (45, 105, 47364)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964701 378143 693646 929430 313029 899433 496098 795963 526077 237146 018667 526209 987688 280525 024309 871776 606310 337850 385260 185832 943382 724379 580627 282060 983769 > 27¹⁰⁵ [i]