MinT - Best Known (58, 109, s)-Nets in Base 27

Best Known (58, 109, s)-Nets in Base 27

(58, 109, 216)-Net over F₂₇ — Constructive and digital

Digital (58, 109, 216)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 21, 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. digital (6, 31, 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]
3. digital (6, 57, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(58, 109, 370)-Net in Base 27 — Constructive

(58, 109, 370)-net in base 27, using

27¹ times duplication [i] based on (57, 108, 370)-net in base 27, using
- t-expansion [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]

(58, 109, 1014)-Net over F₂₇ — Digital

Digital (58, 109, 1014)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 109, 597220)-Net in Base 27 — Upper bound on s

There is no (58, 109, 597221)-net in base 27, because

1 times m-reduction [i] would yield (58, 108, 597221)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38663 433062 920988 831495 356227 864152 608310 631596 054012 447186 531608 327444 368456 103229 452822 335833 144397 482723 555481 865073 885362 713629 520739 405031 041833 162915 > 27¹⁰⁸ [i]