MinT - Best Known (86−31, 86, s)-Nets in Base 27

Best Known (86−31, 86, s)-Nets in Base 27

(86−31, 86, 304)-Net over F₂₇ — Constructive and digital

Digital (55, 86, 304)-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, 16, 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, 21, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
4. digital (6, 37, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(86−31, 86, 486)-Net in Base 27 — Constructive

(55, 86, 486)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (8, 23, 116)-net in base 27, using
  - 1 times m-reduction [i] based on (8, 24, 116)-net in base 27, using
    - base change [i] based on digital (2, 18, 116)-net over F₈₁, using
      - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]
2. (32, 63, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (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]

(86−31, 86, 5891)-Net over F₂₇ — Digital

Digital (55, 86, 5891)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−31, 86, large)-Net in Base 27 — Upper bound on s

There is no (55, 86, large)-net in base 27, because

29 times m-reduction [i] would yield (55, 57, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (86−31, 86, s)-Nets in Base 27

(86−31, 86, 304)-Net over F27 — Constructive and digital

(86−31, 86, 486)-Net in Base 27 — Constructive

(86−31, 86, 5891)-Net over F27 — Digital

(86−31, 86, large)-Net in Base 27 — Upper bound on s

(86−31, 86, 304)-Net over F₂₇ — Constructive and digital

(86−31, 86, 5891)-Net over F₂₇ — Digital