MinT - Best Known (65−45, 65, s)-Nets in Base 27

Best Known (65−45, 65, s)-Nets in Base 27

(65−45, 65, 108)-Net over F₂₇ — Constructive and digital

Digital (20, 65, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 65, 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]

(65−45, 65, 116)-Net in Base 27 — Constructive

(20, 65, 116)-net in base 27, using

7 times m-reduction [i] based on (20, 72, 116)-net in base 27, using
- base change [i] based on digital (2, 54, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(65−45, 65, 148)-Net over F₂₇ — Digital

Digital (20, 65, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 65, 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]

(65−45, 65, 5068)-Net in Base 27 — Upper bound on s

There is no (20, 65, 5069)-net in base 27, because

1 times m-reduction [i] would yield (20, 64, 5069)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 40 577837 393478 916001 732623 760361 485499 128192 770895 681663 854042 000339 229882 023065 797346 775233 > 27⁶⁴ [i]