MinT - Best Known (37, 37+61, s)-Nets in Base 27

Best Known (37, 37+61, s)-Nets in Base 27

(37, 37+61, 116)-Net over F₂₇ — Constructive and digital

Digital (37, 98, 116)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 33, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]
2. digital (4, 65, 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]

(37, 37+61, 172)-Net in Base 27 — Constructive

(37, 98, 172)-net in base 27, using

t-expansion [i] based on (34, 98, 172)-net in base 27, using
- 10 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]

(37, 37+61, 244)-Net over F₂₇ — Digital

Digital (37, 98, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 98, 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+61, 298)-Net in Base 27

(37, 98, 298)-net in base 27, using

2 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+61, 19658)-Net in Base 27 — Upper bound on s

There is no (37, 98, 19659)-net in base 27, because

1 times m-reduction [i] would yield (37, 97, 19659)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 962280 620458 545784 803505 638459 758141 101368 026915 393502 201533 891930 436564 344377 820185 945411 428895 918332 902431 979249 668834 251958 683095 316909 > 27⁹⁷ [i]