MinT - Best Known (89−44, 89, s)-Nets in Base 27

Best Known (89−44, 89, s)-Nets in Base 27

(89−44, 89, 192)-Net over F₂₇ — Constructive and digital

Digital (45, 89, 192)-net over F₂₇, using

2 times m-reduction [i] based on digital (45, 91, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 34, 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, 57, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(89−44, 89, 370)-Net in Base 27 — Constructive

(45, 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]

(89−44, 89, 615)-Net over F₂₇ — Digital

Digital (45, 89, 615)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−44, 89, 214967)-Net in Base 27 — Upper bound on s

There is no (45, 89, 214968)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 624945 934368 795597 119367 275344 625381 063923 483156 183167 886314 340977 127981 190816 406680 312231 433778 774511 704486 427536 459121 475921 > 27⁸⁹ [i]