MinT - Best Known (58, 58+39, s)-Nets in Base 27

Best Known (58, 58+39, s)-Nets in Base 27

(58, 58+39, 264)-Net over F₂₇ — Constructive and digital

Digital (58, 97, 264)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 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 (9, 28, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]
3. digital (10, 49, 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]

(58, 58+39, 434)-Net in Base 27 — Constructive

(58, 97, 434)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. (35, 74, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (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]

(58, 58+39, 2623)-Net over F₂₇ — Digital

Digital (58, 97, 2623)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+39, 5204692)-Net in Base 27 — Upper bound on s

There is no (58, 97, 5204693)-net in base 27, because

1 times m-reduction [i] would yield (58, 96, 5204693)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 257585 642737 686341 423818 201973 549218 339634 138761 504238 310100 835046 045316 889215 261775 114666 235198 978508 841589 589708 933901 939490 817450 599507 > 27⁹⁶ [i]