MinT - Best Known (52, 52+51, s)-Nets in Base 27

Best Known (52, 52+51, s)-Nets in Base 27

(52, 52+51, 196)-Net over F₂₇ — Constructive and digital

Digital (52, 103, 196)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 34, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]
2. digital (18, 69, 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, 52+51, 370)-Net in Base 27 — Constructive

(52, 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]

(52, 52+51, 679)-Net over F₂₇ — Digital

Digital (52, 103, 679)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+51, 270767)-Net in Base 27 — Upper bound on s

There is no (52, 103, 270768)-net in base 27, because

1 times m-reduction [i] would yield (52, 102, 270768)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 802441 746093 121864 195966 488821 318755 859765 943580 920333 071123 632503 602655 714111 493884 489712 631891 825951 698531 729728 825168 206599 529434 155564 690657 > 27¹⁰² [i]