MinT - Best Known (110−52, 110, s)-Nets in Base 27

Best Known (110−52, 110, s)-Nets in Base 27

(110−52, 110, 210)-Net over F₂₇ — Constructive and digital

Digital (58, 110, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 21, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 30, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 59, 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]

(110−52, 110, 370)-Net in Base 27 — Constructive

(58, 110, 370)-net in base 27, using

27² times duplication [i] based on (56, 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]

(110−52, 110, 959)-Net over F₂₇ — Digital

Digital (58, 110, 959)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−52, 110, 461399)-Net in Base 27 — Upper bound on s

There is no (58, 110, 461400)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 28 186148 668927 493204 568356 354955 672191 759794 119460 326031 067690 820797 066692 349116 641208 524398 190469 823726 312264 233671 053862 347500 486495 574795 985977 722110 231057 > 27¹¹⁰ [i]