MinT - Best Known (86−52, 86, s)-Nets in Base 27

Best Known (86−52, 86, s)-Nets in Base 27

(86−52, 86, 128)-Net over F₂₇ — Constructive and digital

Digital (34, 86, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 30, 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, 56, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(86−52, 86, 172)-Net in Base 27 — Constructive

(34, 86, 172)-net in base 27, using

22 times m-reduction [i] based on (34, 108, 172)-net in base 27, using
- base change [i] based on digital (7, 81, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(86−52, 86, 220)-Net over F₂₇ — Digital

Digital (34, 86, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 86, 220)-net over F₂₇, using
- net from sequence [i] based on digital (33, 219)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 33 and N(F) ≥ 220, using
    - Drinfeld modules of rank 1 [i]

(86−52, 86, 298)-Net in Base 27

(34, 86, 298)-net in base 27, using

2 times m-reduction [i] based on (34, 88, 298)-net in base 27, using
- base change [i] based on digital (12, 66, 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]

(86−52, 86, 22007)-Net in Base 27 — Upper bound on s

There is no (34, 86, 22008)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1252 294300 995562 511957 243091 066333 074245 904892 308169 475633 062587 387995 656382 020474 596370 072624 801170 091953 315555 873588 748753 > 27⁸⁶ [i]