MinT - Best Known (107−61, 107, s)-Nets in Base 27

Best Known (107−61, 107, s)-Nets in Base 27

(107−61, 107, 170)-Net over F₂₇ — Constructive and digital

Digital (46, 107, 170)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (9, 70, 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]

(107−61, 107, 305)-Net over F₂₇ — Digital

Digital (46, 107, 305)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(107−61, 107, 370)-Net in Base 27 — Constructive

(46, 107, 370)-net in base 27, using

t-expansion [i] based on (43, 107, 370)-net in base 27, using
- 1 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]

(107−61, 107, 52865)-Net in Base 27 — Upper bound on s

There is no (46, 107, 52866)-net in base 27, because

1 times m-reduction [i] would yield (46, 106, 52866)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 050682 983204 138300 420971 469383 312504 956543 600698 453448 349604 586388 293796 851590 801475 301520 975071 058453 042604 717712 274962 614368 871960 730572 258793 327725 > 27¹⁰⁶ [i]