MinT - Best Known (53, 53+37, s)-Nets in Base 27

Best Known (53, 53+37, s)-Nets in Base 27

(53, 53+37, 252)-Net over F₂₇ — Constructive and digital

Digital (53, 90, 252)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 19, 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]
2. digital (7, 25, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 46, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(53, 53+37, 408)-Net in Base 27 — Constructive

(53, 90, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (34, 71, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
    - base change [i] based on digital (16, 54, 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]

(53, 53+37, 2099)-Net over F₂₇ — Digital

Digital (53, 90, 2099)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+37, 3470996)-Net in Base 27 — Upper bound on s

There is no (53, 90, 3470997)-net in base 27, because

1 times m-reduction [i] would yield (53, 89, 3470997)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 624999 788348 947520 234313 799606 310349 454997 134059 058919 444902 810641 864956 264678 792990 771309 166925 537141 491290 755280 343076 964865 > 27⁸⁹ [i]