MinT - Best Known (49, 49+51, s)-Nets in Base 27

Best Known (49, 49+51, s)-Nets in Base 27

(49, 49+51, 192)-Net over F₂₇ — Constructive and digital

Digital (49, 100, 192)-net over F₂₇, using

3 times m-reduction [i] based on digital (49, 103, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 38, 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, 65, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(49, 49+51, 370)-Net in Base 27 — Constructive

(49, 100, 370)-net in base 27, using

t-expansion [i] based on (43, 100, 370)-net in base 27, using
- 8 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]

(49, 49+51, 552)-Net over F₂₇ — Digital

Digital (49, 100, 552)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+51, 182314)-Net in Base 27 — Upper bound on s

There is no (49, 100, 182315)-net in base 27, because

1 times m-reduction [i] would yield (49, 99, 182315)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5070 139639 022322 524432 478025 368761 670532 895837 377305 651619 589394 418520 219614 830683 309098 622788 188398 828693 973738 383426 033317 658050 460517 183279 > 27⁹⁹ [i]