MinT - Best Known (105−58, 105, s)-Nets in Base 27

Best Known (105−58, 105, s)-Nets in Base 27

(105−58, 105, 178)-Net over F₂₇ — Constructive and digital

Digital (47, 105, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 37, 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, 68, 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]

(105−58, 105, 358)-Net over F₂₇ — Digital

Digital (47, 105, 358)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−58, 105, 370)-Net in Base 27 — Constructive

(47, 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]

(105−58, 105, 68324)-Net in Base 27 — Upper bound on s

There is no (47, 105, 68325)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964643 052286 919828 718238 339010 469273 935454 148949 606019 970146 846369 908853 135384 823190 847854 919763 522234 167592 273635 275413 558289 084021 946308 517683 187859 > 27¹⁰⁵ [i]