MinT - Best Known (95−49, 95, s)-Nets in Base 27

Best Known (95−49, 95, s)-Nets in Base 27

(95−49, 95, 192)-Net over F₂₇ — Constructive and digital

Digital (46, 95, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 35, 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, 60, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(95−49, 95, 370)-Net in Base 27 — Constructive

(46, 95, 370)-net in base 27, using

t-expansion [i] based on (43, 95, 370)-net in base 27, using
- 13 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]

(95−49, 95, 495)-Net over F₂₇ — Digital

Digital (46, 95, 495)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(95−49, 95, 152239)-Net in Base 27 — Upper bound on s

There is no (46, 95, 152240)-net in base 27, because

1 times m-reduction [i] would yield (46, 94, 152240)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 353 342420 508744 351255 482971 452655 885683 120523 101448 203295 455229 586464 973003 886422 639642 029172 507165 912792 623201 470482 388416 213963 783169 > 27⁹⁴ [i]