MinT - Best Known (62−45, 62, s)-Nets in Base 27

Best Known (62−45, 62, s)-Nets in Base 27

(62−45, 62, 96)-Net over F₂₇ — Constructive and digital

Digital (17, 62, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 62, 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]

(62−45, 62, 100)-Net in Base 27 — Constructive

(17, 62, 100)-net in base 27, using

2 times m-reduction [i] based on (17, 64, 100)-net in base 27, using
- base change [i] based on digital (1, 48, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(62−45, 62, 144)-Net over F₂₇ — Digital

Digital (17, 62, 144)-net over F₂₇, using

t-expansion [i] based on digital (16, 62, 144)-net over F₂₇, using
- net from sequence [i] based on digital (16, 143)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 16 and N(F) ≥ 144, using
    - a curve from the manYPoints database [i]

(62−45, 62, 3229)-Net in Base 27 — Upper bound on s

There is no (17, 62, 3230)-net in base 27, because

1 times m-reduction [i] would yield (17, 61, 3230)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2063 381251 293636 379853 363475 661619 018919 446137 044201 248627 589834 475992 997336 100438 161221 > 27⁶¹ [i]