MinT - Best Known (51, 51+41, s)-Nets in Base 27

Best Known (51, 51+41, s)-Nets in Base 27

(51, 51+41, 228)-Net over F₂₇ — Constructive and digital

Digital (51, 92, 228)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 19, 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, 26, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (6, 47, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(51, 51+41, 370)-Net in Base 27 — Constructive

(51, 92, 370)-net in base 27, using

t-expansion [i] based on (43, 92, 370)-net in base 27, using
- 16 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]

(51, 51+41, 1208)-Net over F₂₇ — Digital

Digital (51, 92, 1208)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+41, 1040000)-Net in Base 27 — Upper bound on s

There is no (51, 92, 1040001)-net in base 27, because

1 times m-reduction [i] would yield (51, 91, 1040001)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 17951 603459 232695 358012 902434 126168 158880 167379 248701 489077 234768 061100 925408 068803 044979 475956 643263 571002 136201 439279 557785 889201 > 27⁹¹ [i]