MinT - Best Known (103−58, 103, s)-Nets in Base 27

Best Known (103−58, 103, s)-Nets in Base 27

(103−58, 103, 170)-Net over F₂₇ — Constructive and digital

Digital (45, 103, 170)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 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, 67, 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]

(103−58, 103, 317)-Net over F₂₇ — Digital

Digital (45, 103, 317)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(103−58, 103, 370)-Net in Base 27 — Constructive

(45, 103, 370)-net in base 27, using

t-expansion [i] based on (43, 103, 370)-net in base 27, using
- 5 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]

(103−58, 103, 54429)-Net in Base 27 — Upper bound on s

There is no (45, 103, 54430)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 615520 575972 222600 559712 512712 943543 027591 012524 573805 985181 379884 052632 253433 156941 027493 335243 543896 822889 631099 599218 888342 850961 619502 127989 > 27¹⁰³ [i]