MinT - Best Known (78−30, 78, s)-Nets in Base 27

Best Known (78−30, 78, s)-Nets in Base 27

(78−30, 78, 256)-Net over F₂₇ — Constructive and digital

Digital (48, 78, 256)-net over F₂₇, using

1 times m-reduction [i] based on digital (48, 79, 256)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 11, 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, 14, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (4, 19, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  4. digital (4, 35, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(78−30, 78, 408)-Net in Base 27 — Constructive

(48, 78, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 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, 62, 370)-net in base 27, using
  - 2 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]

(78−30, 78, 3192)-Net over F₂₇ — Digital

Digital (48, 78, 3192)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−30, 78, 6853057)-Net in Base 27 — Upper bound on s

There is no (48, 78, 6853058)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 4429 695997 927362 744640 500129 562029 459721 130767 977547 527660 784782 578191 966315 886398 386736 438910 821579 193690 581705 > 27⁷⁸ [i]