MinT - Best Known (52, 98, s)-Nets in Base 27

Best Known (52, 98, s)-Nets in Base 27

(52, 98, 204)-Net over F₂₇ — Constructive and digital

Digital (52, 98, 204)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 34, 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, 64, 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]

(52, 98, 370)-Net in Base 27 — Constructive

(52, 98, 370)-net in base 27, using

t-expansion [i] based on (43, 98, 370)-net in base 27, using
- 10 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]

(52, 98, 911)-Net over F₂₇ — Digital

Digital (52, 98, 911)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 98, 455329)-Net in Base 27 — Upper bound on s

There is no (52, 98, 455330)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 187 783482 881521 699857 484431 321192 729864 764257 883934 267890 148542 347960 809083 044404 563549 203105 734809 800809 202084 057847 417064 563457 385202 328441 > 27⁹⁸ [i]