MinT - Best Known (84−48, 84, s)-Nets in Base 27

Best Known (84−48, 84, s)-Nets in Base 27

(84−48, 84, 152)-Net over F₂₇ — Constructive and digital

Digital (36, 84, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 30, 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]
2. digital (6, 54, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(84−48, 84, 224)-Net in Base 27 — Constructive

(36, 84, 224)-net in base 27, using

8 times m-reduction [i] based on (36, 92, 224)-net in base 27, using
- base change [i] based on digital (13, 69, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(84−48, 84, 248)-Net over F₂₇ — Digital

Digital (36, 84, 248)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−48, 84, 298)-Net in Base 27

(36, 84, 298)-net in base 27, using

12 times m-reduction [i] based on (36, 96, 298)-net in base 27, using
- base change [i] based on digital (12, 72, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(84−48, 84, 38549)-Net in Base 27 — Upper bound on s

There is no (36, 84, 38550)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 716318 949394 660413 772196 689052 012309 696399 233910 772619 474892 624611 086458 311848 246476 097331 543606 828217 509361 626370 857841 > 27⁸⁴ [i]