MinT - Best Known (61−41, 61, s)-Nets in Base 27

Best Known (61−41, 61, s)-Nets in Base 27

(61−41, 61, 108)-Net over F₂₇ — Constructive and digital

Digital (20, 61, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 61, 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]

(61−41, 61, 148)-Net over F₂₇ — Digital

Digital (20, 61, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 61, 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]

(61−41, 61, 150)-Net in Base 27 — Constructive

(20, 61, 150)-net in base 27, using

3 times m-reduction [i] based on (20, 64, 150)-net in base 27, using
- base change [i] based on digital (4, 48, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(61−41, 61, 154)-Net in Base 27

(20, 61, 154)-net in base 27, using

3 times m-reduction [i] based on (20, 64, 154)-net in base 27, using
- base change [i] based on digital (4, 48, 154)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 153)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 154, using
      - a curve from the manYPoints database [i]

(61−41, 61, 6276)-Net in Base 27 — Upper bound on s

There is no (20, 61, 6277)-net in base 27, because

1 times m-reduction [i] would yield (20, 60, 6277)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 76 309425 189065 255027 180772 126124 756468 727992 012326 219810 695658 711874 228884 319644 937521 > 27⁶⁰ [i]