MinT - Best Known (58, 58+38, s)-Nets in Base 27

Best Known (58, 58+38, s)-Nets in Base 27

(58, 58+38, 270)-Net over F₂₇ — Constructive and digital

Digital (58, 96, 270)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 19, 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 (10, 29, 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, 48, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(58, 58+38, 438)-Net in Base 27 — Constructive

(58, 96, 438)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (5, 24, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]
2. (34, 72, 370)-net in base 27, using
  - base change [i] based on digital (16, 54, 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+38, 2934)-Net over F₂₇ — Digital

Digital (58, 96, 2934)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+38, 5204692)-Net in Base 27 — Upper bound on s

There is no (58, 96, 5204693)-net in base 27, because

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]