MinT - Best Known (56, 56+51, s)-Nets in Base 27

Best Known (56, 56+51, s)-Nets in Base 27

(56, 56+51, 204)-Net over F₂₇ — Constructive and digital

Digital (56, 107, 204)-net over F₂₇, using

3 times m-reduction [i] based on digital (56, 110, 204)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 38, 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, 72, 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]

(56, 56+51, 370)-Net in Base 27 — Constructive

(56, 107, 370)-net in base 27, using

t-expansion [i] based on (43, 107, 370)-net in base 27, using
- 1 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]

(56, 56+51, 892)-Net over F₂₇ — Digital

Digital (56, 107, 892)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+51, 458799)-Net in Base 27 — Upper bound on s

There is no (56, 107, 458800)-net in base 27, because

1 times m-reduction [i] would yield (56, 106, 458800)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 037125 199091 562532 982464 474161 182595 090026 454880 956032 170111 866797 756350 409495 340799 503890 790179 011150 177184 489610 094273 054792 499161 881374 763349 712353 > 27¹⁰⁶ [i]