MinT - Best Known (110−66, 110, s)-Nets in Base 27

Best Known (110−66, 110, s)-Nets in Base 27

(110−66, 110, 146)-Net over F₂₇ — Constructive and digital

Digital (44, 110, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 37, 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 (7, 73, 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]

(110−66, 110, 224)-Net in Base 27 — Constructive

(44, 110, 224)-net in base 27, using

27² times duplication [i] based on (42, 108, 224)-net in base 27, using
- t-expansion [i] based on (40, 108, 224)-net in base 27, using
  - base change [i] based on digital (13, 81, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(110−66, 110, 280)-Net over F₂₇ — Digital

Digital (44, 110, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 110, 280)-net over F₂₇, using
- net from sequence [i] based on digital (42, 279)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
    - Drinfeld modules of rank 1 [i]

(110−66, 110, 298)-Net in Base 27

(44, 110, 298)-net in base 27, using

27² times duplication [i] based on (42, 108, 298)-net in base 27, using
- t-expansion [i] based on (39, 108, 298)-net in base 27, using
  - base change [i] based on digital (12, 81, 298)-net over F₈₁, using
    - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
        a curve from the manYPoints database [i]

(110−66, 110, 29877)-Net in Base 27 — Upper bound on s

There is no (44, 110, 29878)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 28 204292 251371 013611 406453 286661 956275 673435 887911 306826 464730 500451 408314 951440 506887 385979 979899 302450 729977 939094 744078 295524 742534 490876 677501 655786 579517 > 27¹¹⁰ [i]