MinT - Best Known (110−58, 110, s)-Nets in Base 27

Best Known (110−58, 110, s)-Nets in Base 27

(110−58, 110, 192)-Net over F₂₇ — Constructive and digital

Digital (52, 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)

(110−58, 110, 370)-Net in Base 27 — Constructive

(52, 110, 370)-net in base 27, using

27² times duplication [i] based on (50, 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]

(110−58, 110, 487)-Net over F₂₇ — Digital

Digital (52, 110, 487)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−58, 110, 120616)-Net in Base 27 — Upper bound on s

There is no (52, 110, 120617)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 28 191201 850766 080470 431858 741824 159914 383222 446400 685270 305268 145135 092546 637932 814440 823571 261893 893042 131311 793655 467130 041160 769479 564142 966850 055588 350043 > 27¹¹⁰ [i]