MinT - Best Known (61, 61+41, s)-Nets in Base 27

Best Known (61, 61+41, s)-Nets in Base 27

(61, 61+41, 272)-Net over F₂₇ — Constructive and digital

Digital (61, 102, 272)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (8, 21, 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, 30, 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]
3. digital (10, 51, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(61, 61+41, 446)-Net in Base 27 — Constructive

(61, 102, 446)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (6, 26, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. (35, 76, 370)-net in base 27, using
  - base change [i] based on digital (16, 57, 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]

(61, 61+41, 2729)-Net over F₂₇ — Digital

Digital (61, 102, 2729)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+41, 5404046)-Net in Base 27 — Upper bound on s

There is no (61, 102, 5404047)-net in base 27, because

1 times m-reduction [i] would yield (61, 101, 5404047)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 696078 600415 968166 503766 970118 873066 191243 571624 195547 548902 824057 989778 397414 192201 655314 650968 874103 353243 947587 517793 616698 639504 707585 363961 > 27¹⁰¹ [i]