MinT - Best Known (56, 56+47, s)-Nets in Base 27

Best Known (56, 56+47, s)-Nets in Base 27

(56, 56+47, 228)-Net over F₂₇ — Constructive and digital

Digital (56, 103, 228)-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, 29, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (6, 53, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(56, 56+47, 370)-Net in Base 27 — Constructive

(56, 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]

(56, 56+47, 1133)-Net over F₂₇ — Digital

Digital (56, 103, 1133)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+47, 807725)-Net in Base 27 — Upper bound on s

There is no (56, 103, 807726)-net in base 27, because

1 times m-reduction [i] would yield (56, 102, 807726)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 795965 964636 450931 182113 748373 970324 688090 120433 357035 871854 864125 561073 819077 251963 314482 921130 976612 600548 899659 785268 234022 838580 056237 728553 > 27¹⁰² [i]