MinT - Best Known (105−57, 105, s)-Nets in Base 27

Best Known (105−57, 105, s)-Nets in Base 27

(105−57, 105, 188)-Net over F₂₇ — Constructive and digital

Digital (48, 105, 188)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 38, 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]
2. digital (10, 67, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(105−57, 105, 370)-Net in Base 27 — Constructive

(48, 105, 370)-net in base 27, using

t-expansion [i] based on (43, 105, 370)-net in base 27, using
- 3 times m-reduction [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]

(105−57, 105, 395)-Net over F₂₇ — Digital

Digital (48, 105, 395)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−57, 105, 90044)-Net in Base 27 — Upper bound on s

There is no (48, 105, 90045)-net in base 27, because

1 times m-reduction [i] would yield (48, 104, 90045)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72769 621713 321247 721429 078911 020019 070863 177058 883028 771881 400923 674141 389764 099983 094182 811185 092673 611581 872511 710166 143666 175716 241150 552852 512273 > 27¹⁰⁴ [i]