MinT - Best Known (31, 64, s)-Nets in Base 27

Best Known (31, 64, s)-Nets in Base 27

(31, 64, 166)-Net over F₂₇ — Constructive and digital

Digital (31, 64, 166)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 23, 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, 41, 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]

(31, 64, 224)-Net in Base 27 — Constructive

(31, 64, 224)-net in base 27, using

8 times m-reduction [i] based on (31, 72, 224)-net in base 27, using
- base change [i] based on digital (13, 54, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(31, 64, 374)-Net over F₂₇ — Digital

Digital (31, 64, 374)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 64, 113117)-Net in Base 27 — Upper bound on s

There is no (31, 64, 113118)-net in base 27, because

1 times m-reduction [i] would yield (31, 63, 113118)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 499546 553645 090994 064092 745364 885937 006874 973013 848309 030765 506165 603117 345721 987618 094497 > 27⁶³ [i]