MinT - Best Known (109−63, 109, s)-Nets in Base 27

Best Known (109−63, 109, s)-Nets in Base 27

(109−63, 109, 166)-Net over F₂₇ — Constructive and digital

Digital (46, 109, 166)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 38, 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 (8, 71, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(109−63, 109, 288)-Net over F₂₇ — Digital

Digital (46, 109, 288)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−63, 109, 370)-Net in Base 27 — Constructive

(46, 109, 370)-net in base 27, using

27¹ times duplication [i] based on (45, 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]

(109−63, 109, 46301)-Net in Base 27 — Upper bound on s

There is no (46, 109, 46302)-net in base 27, because

1 times m-reduction [i] would yield (46, 108, 46302)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38676 868016 988337 021465 437811 084850 856653 787609 939582 340778 232828 283900 298393 728788 779235 063730 818063 121115 764175 384314 323194 485214 014952 055616 140470 632729 > 27¹⁰⁸ [i]