MinT - Best Known (53, 53+40, s)-Nets in Base 27

Best Known (53, 53+40, s)-Nets in Base 27

(53, 53+40, 240)-Net over F₂₇ — Constructive and digital

Digital (53, 93, 240)-net over F₂₇, using

1 times m-reduction [i] based on digital (53, 94, 240)-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 (7, 27, 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]
  3. digital (7, 48, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(53, 53+40, 370)-Net in Base 27 — Constructive

(53, 93, 370)-net in base 27, using

t-expansion [i] based on (43, 93, 370)-net in base 27, using
- 15 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]

(53, 53+40, 1553)-Net over F₂₇ — Digital

Digital (53, 93, 1553)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+40, 1446010)-Net in Base 27 — Upper bound on s

There is no (53, 93, 1446011)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 086875 219223 889316 788071 246594 821291 325016 979088 303185 641733 597556 633402 156685 348428 456580 987977 355776 669586 405925 106885 270586 537689 > 27⁹³ [i]