MinT - Best Known (108−53, 108, s)-Nets in Base 27

Best Known (108−53, 108, s)-Nets in Base 27

(108−53, 108, 204)-Net over F₂₇ — Constructive and digital

Digital (55, 108, 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, 71, 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]

(108−53, 108, 370)-Net in Base 27 — Constructive

(55, 108, 370)-net in base 27, using

t-expansion [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]

(108−53, 108, 746)-Net over F₂₇ — Digital

Digital (55, 108, 746)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−53, 108, 315437)-Net in Base 27 — Upper bound on s

There is no (55, 108, 315438)-net in base 27, because

1 times m-reduction [i] would yield (55, 107, 315438)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1431 975323 377570 458064 850384 120892 863593 110932 825866 412987 961321 022223 899808 020199 125574 797239 091881 426934 288351 738450 568833 250665 033354 228996 179348 311181 > 27¹⁰⁷ [i]