MinT - Best Known (55, 55+49, s)-Nets in Base 27

Best Known (55, 55+49, s)-Nets in Base 27

(55, 55+49, 210)-Net over F₂₇ — Constructive and digital

Digital (55, 104, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 20, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 28, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 56, 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]

(55, 55+49, 370)-Net in Base 27 — Constructive

(55, 104, 370)-net in base 27, using

t-expansion [i] based on (43, 104, 370)-net in base 27, using
- 4 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]

(55, 55+49, 934)-Net over F₂₇ — Digital

Digital (55, 104, 934)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+49, 523980)-Net in Base 27 — Upper bound on s

There is no (55, 104, 523981)-net in base 27, because

1 times m-reduction [i] would yield (55, 103, 523981)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 478624 878138 615586 843289 146116 313257 708220 086416 331631 989542 366686 934078 803695 606270 211397 936796 258942 916765 204877 540291 056188 333299 348272 862433 > 27¹⁰³ [i]