MinT - Best Known (55, 106, s)-Nets in Base 27

Best Known (55, 106, s)-Nets in Base 27

(55, 106, 204)-Net over F₂₇ — Constructive and digital

Digital (55, 106, 204)-net over F₂₇, using

1 times m-reduction [i] based on digital (55, 107, 204)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 37, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]
  2. digital (18, 70, 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]

(55, 106, 370)-Net in Base 27 — Constructive

(55, 106, 370)-net in base 27, using

t-expansion [i] based on (43, 106, 370)-net in base 27, using
- 2 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(55, 106, 835)-Net over F₂₇ — Digital

Digital (55, 106, 835)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 106, 402129)-Net in Base 27 — Upper bound on s

There is no (55, 106, 402130)-net in base 27, because

1 times m-reduction [i] would yield (55, 105, 402130)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964263 812608 293317 825760 345790 602420 135175 882342 027843 226301 102355 550840 939091 361873 513719 368936 338754 406608 555502 234145 282994 655099 247297 249649 510117 > 27¹⁰⁵ [i]