MinT - Best Known (56, 56+45, s)-Nets in Base 27

Best Known (56, 56+45, s)-Nets in Base 27

(56, 56+45, 234)-Net over F₂₇ — Constructive and digital

Digital (56, 101, 234)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 21, 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, 28, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (7, 52, 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]

(56, 56+45, 370)-Net in Base 27 — Constructive

(56, 101, 370)-net in base 27, using

t-expansion [i] based on (43, 101, 370)-net in base 27, using
- 7 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]

(56, 56+45, 1303)-Net over F₂₇ — Digital

Digital (56, 101, 1303)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+45, 1117054)-Net in Base 27 — Upper bound on s

There is no (56, 101, 1117055)-net in base 27, because

1 times m-reduction [i] would yield (56, 100, 1117055)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136893 982853 609866 902421 928402 584655 560456 849762 089584 798814 407115 211233 419273 790888 330149 171601 995440 766052 630159 951277 212517 644992 486555 490501 > 27¹⁰⁰ [i]