MinT - Best Known (57−37, 57, s)-Nets in Base 27

Best Known (57−37, 57, s)-Nets in Base 27

(57−37, 57, 108)-Net over F₂₇ — Constructive and digital

Digital (20, 57, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 57, 108)-net over F₂₇, using
- net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
    - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(57−37, 57, 148)-Net over F₂₇ — Digital

Digital (20, 57, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 57, 148)-net over F₂₇, using
- net from sequence [i] based on digital (18, 147)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 148, using
    - a curve from the manYPoints database [i]

(57−37, 57, 160)-Net in Base 27 — Constructive

(20, 57, 160)-net in base 27, using

3 times m-reduction [i] based on (20, 60, 160)-net in base 27, using
- base change [i] based on digital (5, 45, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(57−37, 57, 167)-Net in Base 27

(20, 57, 167)-net in base 27, using

3 times m-reduction [i] based on (20, 60, 167)-net in base 27, using
- base change [i] based on digital (5, 45, 167)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
      - a curve from the manYPoints database [i]

(57−37, 57, 8237)-Net in Base 27 — Upper bound on s

There is no (20, 57, 8238)-net in base 27, because

1 times m-reduction [i] would yield (20, 56, 8238)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 143 445583 787426 051882 619491 524797 479820 267498 002593 814804 090622 703056 103159 076349 > 27⁵⁶ [i]