MinT - Best Known (58, 58+48, s)-Nets in Base 27

Best Known (58, 58+48, s)-Nets in Base 27

(58, 58+48, 228)-Net over F₂₇ — Constructive and digital

Digital (58, 106, 228)-net over F₂₇, using

1 times m-reduction [i] based on digital (58, 107, 228)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (6, 22, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]
  2. digital (6, 30, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
  3. digital (6, 55, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(58, 58+48, 370)-Net in Base 27 — Constructive

(58, 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]

(58, 58+48, 1218)-Net over F₂₇ — Digital

Digital (58, 106, 1218)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+48, 791114)-Net in Base 27 — Upper bound on s

There is no (58, 106, 791115)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 035810 833504 340485 121954 844104 912534 553120 842279 404668 184175 982948 696653 019809 680870 464620 733810 408414 517655 720324 299917 284462 723940 716382 371272 727569 > 27¹⁰⁶ [i]