MinT - Best Known (37, 37+51, s)-Nets in Base 27

Best Known (37, 37+51, s)-Nets in Base 27

(37, 37+51, 152)-Net over F₂₇ — Constructive and digital

Digital (37, 88, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 31, 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, 57, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(37, 37+51, 224)-Net in Base 27 — Constructive

(37, 88, 224)-net in base 27, using

8 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+51, 244)-Net over F₂₇ — Digital

Digital (37, 88, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 88, 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+51, 298)-Net in Base 27

(37, 88, 298)-net in base 27, using

12 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+51, 37466)-Net in Base 27 — Upper bound on s

There is no (37, 88, 37467)-net in base 27, because

1 times m-reduction [i] would yield (37, 87, 37467)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 33780 505220 216003 307452 466107 936760 235841 918443 122514 713705 711324 351756 097595 824449 059641 102106 713807 910115 547709 775941 913615 > 27⁸⁷ [i]