MinT - Best Known (50, 50+39, s)-Nets in Base 27

Best Known (50, 50+39, s)-Nets in Base 27

(50, 50+39, 228)-Net over F₂₇ — Constructive and digital

Digital (50, 89, 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, 25, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (6, 45, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(50, 50+39, 370)-Net in Base 27 — Constructive

(50, 89, 370)-net in base 27, using

t-expansion [i] based on (43, 89, 370)-net in base 27, using
- 19 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]

(50, 50+39, 1320)-Net over F₂₇ — Digital

Digital (50, 89, 1320)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+39, 1299311)-Net in Base 27 — Upper bound on s

There is no (50, 89, 1299312)-net in base 27, because

1 times m-reduction [i] would yield (50, 88, 1299312)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 912045 127030 684295 303702 204404 970381 925359 203830 854997 476770 295479 282037 308812 924230 021076 206027 844880 612839 863411 368729 498689 > 27⁸⁸ [i]