MinT - Best Known (110−63, 110, s)-Nets in Base 27

Best Known (110−63, 110, s)-Nets in Base 27

(110−63, 110, 170)-Net over F₂₇ — Constructive and digital

Digital (47, 110, 170)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 38, 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 (9, 72, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(110−63, 110, 305)-Net over F₂₇ — Digital

Digital (47, 110, 305)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−63, 110, 370)-Net in Base 27 — Constructive

(47, 110, 370)-net in base 27, using

27² times duplication [i] based on (45, 108, 370)-net in base 27, using
- t-expansion [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]

(110−63, 110, 51497)-Net in Base 27 — Upper bound on s

There is no (47, 110, 51498)-net in base 27, because

1 times m-reduction [i] would yield (47, 109, 51498)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 044444 730564 652198 185867 320743 732455 713375 452931 566433 248729 111285 000698 208703 894458 708736 518764 447651 599158 554416 819979 164537 464934 485635 293480 773415 098569 > 27¹⁰⁹ [i]