MinT - Best Known (54, 54+49, s)-Nets in Base 27

Best Known (54, 54+49, s)-Nets in Base 27

(54, 54+49, 204)-Net over F₂₇ — Constructive and digital

Digital (54, 103, 204)-net over F₂₇, using

1 times m-reduction [i] based on digital (54, 104, 204)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 36, 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, 68, 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]

(54, 54+49, 370)-Net in Base 27 — Constructive

(54, 103, 370)-net in base 27, using

t-expansion [i] based on (43, 103, 370)-net in base 27, using
- 5 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]

(54, 54+49, 874)-Net over F₂₇ — Digital

Digital (54, 103, 874)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+49, 456744)-Net in Base 27 — Upper bound on s

There is no (54, 103, 456745)-net in base 27, because

1 times m-reduction [i] would yield (54, 102, 456745)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 795145 970848 046221 943362 981016 379429 078355 716681 380156 247874 097388 027173 821151 489581 637107 365436 717999 957367 998882 016945 984585 029485 001195 062177 > 27¹⁰² [i]