MinT - Best Known (53, 53+57, s)-Nets in Base 27

Best Known (53, 53+57, s)-Nets in Base 27

(53, 53+57, 192)-Net over F₂₇ — Constructive and digital

Digital (53, 110, 192)-net over F₂₇, using

t-expansion [i] based on digital (51, 110, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]
  2. digital (11, 70, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(53, 53+57, 370)-Net in Base 27 — Constructive

(53, 110, 370)-net in base 27, using

27² times duplication [i] based on (51, 108, 370)-net in base 27, using
- t-expansion [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+57, 540)-Net over F₂₇ — Digital

Digital (53, 110, 540)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+57, 162213)-Net in Base 27 — Upper bound on s

There is no (53, 110, 162214)-net in base 27, because

1 times m-reduction [i] would yield (53, 109, 162214)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 043948 172078 134047 667305 674705 048054 039846 916574 280025 618253 751576 848495 011228 716568 109522 660150 309350 335989 454290 513573 089541 733015 078711 457860 916789 787385 > 27¹⁰⁹ [i]