MinT - Best Known (95−44, 95, s)-Nets in Base 27

Best Known (95−44, 95, s)-Nets in Base 27

(95−44, 95, 210)-Net over F₂₇ — Constructive and digital

Digital (51, 95, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 18, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 26, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 51, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(95−44, 95, 370)-Net in Base 27 — Constructive

(51, 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]

(95−44, 95, 965)-Net over F₂₇ — Digital

Digital (51, 95, 965)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(95−44, 95, 528152)-Net in Base 27 — Upper bound on s

There is no (51, 95, 528153)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 9540 405579 973288 136797 750443 527515 740162 108474 149964 525560 394064 369796 434663 964762 093035 974173 906181 242341 934716 013382 320080 994297 916153 > 27⁹⁵ [i]