MinT - Best Known (59, 105, s)-Nets in Base 27

Best Known (59, 105, s)-Nets in Base 27

(59, 105, 246)-Net over F₂₇ — Constructive and digital

Digital (59, 105, 246)-net over F₂₇, using

1 times m-reduction [i] based on digital (59, 106, 246)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 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 (7, 30, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
  3. digital (7, 54, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(59, 105, 370)-Net in Base 27 — Constructive

(59, 105, 370)-net in base 27, using

t-expansion [i] based on (43, 105, 370)-net in base 27, using
- 3 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]

(59, 105, 1505)-Net over F₂₇ — Digital

Digital (59, 105, 1505)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 105, 1241553)-Net in Base 27 — Upper bound on s

There is no (59, 105, 1241554)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964252 433565 689166 626075 664002 366973 525008 220864 399615 110547 495122 207648 709499 245616 992169 189091 220893 073464 415514 091581 940882 307530 238224 976413 240633 > 27¹⁰⁵ [i]