MinT - Best Known (35, 35+23, s)-Nets in Base 27

Best Known (35, 35+23, s)-Nets in Base 27

(35, 35+23, 228)-Net over F₂₇ — Constructive and digital

Digital (35, 58, 228)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 12, 76)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 730)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁴, 730, F₂₇, 3, 3) (dual of [(730, 3), 2186, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(27⁴, 730, F₂₇, 2, 3) (dual of [(730, 2), 1456, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁴, 730, F₂₇, 3) (dual of [730, 726, 4]-code or 730-cap in PG(3,27)), using
        ovoid in PG(3,Â 27) [i]
    2. digital (1, 8, 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. digital (6, 17, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
3. digital (6, 29, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(35, 35+23, 370)-Net in Base 27 — Constructive

(35, 58, 370)-net in base 27, using

18 times m-reduction [i] based on (35, 76, 370)-net in base 27, using
- base change [i] based on digital (16, 57, 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]

(35, 35+23, 2079)-Net over F₂₇ — Digital

Digital (35, 58, 2079)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(35, 35+23, 4932954)-Net in Base 27 — Upper bound on s

There is no (35, 58, 4932955)-net in base 27, because

1 times m-reduction [i] would yield (35, 57, 4932955)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3870 214647 822151 584857 860909 666472 843738 180394 746371 658097 415578 859383 707944 583379 > 27⁵⁷ [i]