MinT - Best Known (97−49, 97, s)-Nets in Base 27

Best Known (97−49, 97, s)-Nets in Base 27

(97−49, 97, 192)-Net over F₂₇ — Constructive and digital

Digital (48, 97, 192)-net over F₂₇, using

3 times m-reduction [i] based on digital (48, 100, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 37, 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, 63, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(97−49, 97, 370)-Net in Base 27 — Constructive

(48, 97, 370)-net in base 27, using

t-expansion [i] based on (43, 97, 370)-net in base 27, using
- 11 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]

(97−49, 97, 572)-Net over F₂₇ — Digital

Digital (48, 97, 572)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(97−49, 97, 200362)-Net in Base 27 — Upper bound on s

There is no (48, 97, 200363)-net in base 27, because

1 times m-reduction [i] would yield (48, 96, 200363)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 257585 568908 840140 935419 958898 687179 339449 529416 393113 302946 403302 686777 336034 830848 643379 771395 163521 778507 588248 895337 766361 865875 331601 > 27⁹⁶ [i]