MinT - Best Known (88−40, 88, s)-Nets in Base 27

Best Known (88−40, 88, s)-Nets in Base 27

(88−40, 88, 210)-Net over F₂₇ — Constructive and digital

Digital (48, 88, 210)-net over F₂₇, using

1 times m-reduction [i] based on digital (48, 89, 210)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 17, 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, 24, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (7, 48, 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]

(88−40, 88, 370)-Net in Base 27 — Constructive

(48, 88, 370)-net in base 27, using

t-expansion [i] based on (43, 88, 370)-net in base 27, using
- 20 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]

(88−40, 88, 1025)-Net over F₂₇ — Digital

Digital (48, 88, 1025)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−40, 88, 634346)-Net in Base 27 — Upper bound on s

There is no (48, 88, 634347)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 912054 436116 858971 810253 729774 168592 047094 094942 913705 676360 348825 586380 600845 719990 493323 204761 743978 832839 838805 417280 752601 > 27⁸⁸ [i]