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

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

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

Digital (49, 98, 192)-net over F₂₇, using

5 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+49, 370)-Net in Base 27 — Constructive

(49, 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]

(49, 49+49, 615)-Net over F₂₇ — Digital

Digital (49, 98, 615)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+49, 229858)-Net in Base 27 — Upper bound on s

There is no (49, 98, 229859)-net in base 27, because

1 times m-reduction [i] would yield (49, 97, 229859)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 954954 623690 124091 634015 144112 462839 417827 710570 881358 290822 091980 219269 991023 758252 321728 852128 120802 151382 305793 063826 203118 881175 930001 > 27⁹⁷ [i]