MinT - Best Known (81−32, 81, s)-Nets in Base 27

Best Known (81−32, 81, s)-Nets in Base 27

(81−32, 81, 252)-Net over F₂₇ — Constructive and digital

Digital (49, 81, 252)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 17, 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]
2. digital (7, 23, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 41, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(81−32, 81, 408)-Net in Base 27 — Constructive

(49, 81, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (32, 64, 370)-net in base 27, using
  - base change [i] based on digital (16, 48, 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]

(81−32, 81, 2640)-Net over F₂₇ — Digital

Digital (49, 81, 2640)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−32, 81, 4611515)-Net in Base 27 — Upper bound on s

There is no (49, 81, 4611516)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 189807 503953 775698 308079 541463 990321 606047 235513 643243 511079 297797 276834 554128 091329 486059 718227 045280 939648 947841 > 27⁸¹ [i]