MinT - Best Known (37, 37+58, s)-Nets in Base 27

Best Known (37, 37+58, s)-Nets in Base 27

(37, 37+58, 128)-Net over F₂₇ — Constructive and digital

Digital (37, 95, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 33, 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, 62, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(37, 37+58, 224)-Net in Base 27 — Constructive

(37, 95, 224)-net in base 27, using

1 times m-reduction [i] based on (37, 96, 224)-net in base 27, using
- base change [i] based on digital (13, 72, 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]

(37, 37+58, 244)-Net over F₂₇ — Digital

Digital (37, 95, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 95, 244)-net over F₂₇, using
- net from sequence [i] based on digital (36, 243)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 36 and N(F) ≥ 244, using
    - Drinfeld modules of rank 1 [i]

(37, 37+58, 298)-Net in Base 27

(37, 95, 298)-net in base 27, using

5 times m-reduction [i] based on (37, 100, 298)-net in base 27, using
- base change [i] based on digital (12, 75, 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]

(37, 37+58, 21917)-Net in Base 27 — Upper bound on s

There is no (37, 95, 21918)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 9541 052976 113110 928684 128476 577025 727472 918566 108917 274519 929377 241619 794777 303306 500662 320720 979144 160057 344988 220742 336454 422739 017077 > 27⁹⁵ [i]