MinT - Best Known (46, 46+57, s)-Nets in Base 27

Best Known (46, 46+57, s)-Nets in Base 27

(46, 46+57, 178)-Net over F₂₇ — Constructive and digital

Digital (46, 103, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 36, 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]
2. digital (10, 67, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(46, 46+57, 349)-Net over F₂₇ — Digital

Digital (46, 103, 349)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(46, 46+57, 370)-Net in Base 27 — Constructive

(46, 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]

(46, 46+57, 71153)-Net in Base 27 — Upper bound on s

There is no (46, 103, 71154)-net in base 27, because

1 times m-reduction [i] would yield (46, 102, 71154)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 812445 723557 512896 116447 318133 929332 202362 962113 714763 216800 903771 204776 449556 836162 985048 408414 374582 923914 636023 152109 248858 005647 067109 438489 > 27¹⁰² [i]