MinT - Best Known (36, 36+57, s)-Nets in Base 27

Best Known (36, 36+57, s)-Nets in Base 27

(36, 36+57, 128)-Net over F₂₇ — Constructive and digital

Digital (36, 93, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 32, 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, 61, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(36, 36+57, 172)-Net in Base 27 — Constructive

(36, 93, 172)-net in base 27, using

t-expansion [i] based on (34, 93, 172)-net in base 27, using
- 15 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]

(36, 36+57, 244)-Net over F₂₇ — Digital

Digital (36, 93, 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]

(36, 36+57, 298)-Net in Base 27

(36, 93, 298)-net in base 27, using

3 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]

(36, 36+57, 21917)-Net in Base 27 — Upper bound on s

There is no (36, 93, 21918)-net in base 27, because

1 times m-reduction [i] would yield (36, 92, 21918)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484866 555146 423056 668169 662114 907570 059000 676251 377982 066351 955601 002291 567538 781969 354728 540106 264794 277600 964724 565255 384313 120889 > 27⁹² [i]