MinT - Best Known (98−51, 98, s)-Nets in Base 27

Best Known (98−51, 98, s)-Nets in Base 27

(98−51, 98, 192)-Net over F₂₇ — Constructive and digital

Digital (47, 98, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 36, 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, 62, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(98−51, 98, 370)-Net in Base 27 — Constructive

(47, 98, 370)-net in base 27, using

t-expansion [i] based on (43, 98, 370)-net in base 27, using
- 10 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]

(98−51, 98, 480)-Net over F₂₇ — Digital

Digital (47, 98, 480)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(98−51, 98, 140056)-Net in Base 27 — Upper bound on s

There is no (47, 98, 140057)-net in base 27, because

1 times m-reduction [i] would yield (47, 97, 140057)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 955257 838676 738482 748724 478487 255349 679021 487834 760015 738384 012194 075378 944539 922504 018680 393971 203670 745779 344788 161609 022064 325175 179499 > 27⁹⁷ [i]