MinT - Best Known (105−49, 105, s)-Nets in Base 27

Best Known (105−49, 105, s)-Nets in Base 27

(105−49, 105, 216)-Net over F₂₇ — Constructive and digital

Digital (56, 105, 216)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 20, 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, 30, 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, 55, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(105−49, 105, 370)-Net in Base 27 — Constructive

(56, 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]

(105−49, 105, 999)-Net over F₂₇ — Digital

Digital (56, 105, 999)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−49, 105, 601113)-Net in Base 27 — Upper bound on s

There is no (56, 105, 601114)-net in base 27, because

1 times m-reduction [i] would yield (56, 104, 601114)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72750 221036 756683 296056 171301 174014 379290 840643 232244 392812 058363 180077 676235 601801 067188 234897 311906 217994 081000 187327 260017 032211 271193 726138 195313 > 27¹⁰⁴ [i]