MinT - Best Known (89−45, 89, s)-Nets in Base 27

Best Known (89−45, 89, s)-Nets in Base 27

(89−45, 89, 192)-Net over F₂₇ — Constructive and digital

Digital (44, 89, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 33, 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, 56, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(89−45, 89, 370)-Net in Base 27 — Constructive

(44, 89, 370)-net in base 27, using

t-expansion [i] based on (43, 89, 370)-net in base 27, using
- 19 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]

(89−45, 89, 533)-Net over F₂₇ — Digital

Digital (44, 89, 533)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−45, 89, 185058)-Net in Base 27 — Upper bound on s

There is no (44, 89, 185059)-net in base 27, because

1 times m-reduction [i] would yield (44, 88, 185059)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 912131 542918 421827 586264 590776 364067 407293 292805 696723 771362 284525 566661 954423 141411 832621 004366 071092 620023 751873 384253 902637 > 27⁸⁸ [i]