MinT - Best Known (57, 110, s)-Nets in Base 27

Best Known (57, 110, s)-Nets in Base 27

(57, 110, 204)-Net over F₂₇ — Constructive and digital

Digital (57, 110, 204)-net over F₂₇, using

t-expansion [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]

(57, 110, 370)-Net in Base 27 — Constructive

(57, 110, 370)-net in base 27, using

27² times duplication [i] based on (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]

(57, 110, 852)-Net over F₂₇ — Digital

Digital (57, 110, 852)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 110, 406464)-Net in Base 27 — Upper bound on s

There is no (57, 110, 406465)-net in base 27, because

1 times m-reduction [i] would yield (57, 109, 406465)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 043914 856562 530893 254745 586840 257432 496504 024960 938111 176787 669051 429842 270773 982874 543009 780536 019342 448417 919769 062162 120909 914508 223644 278487 919669 545465 > 27¹⁰⁹ [i]