MinT - Best Known (36, 36+25, s)-Nets in Base 27

Best Known (36, 36+25, s)-Nets in Base 27

(36, 36+25, 216)-Net over F₂₇ — Constructive and digital

Digital (36, 61, 216)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 12, 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, 18, 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, 31, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(36, 36+25, 370)-Net in Base 27 — Constructive

(36, 61, 370)-net in base 27, using

19 times m-reduction [i] based on (36, 80, 370)-net in base 27, using
- base change [i] based on digital (16, 60, 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]

(36, 36+25, 1650)-Net over F₂₇ — Digital

Digital (36, 61, 1650)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(36, 36+25, 2918811)-Net in Base 27 — Upper bound on s

There is no (36, 61, 2918812)-net in base 27, because

1 times m-reduction [i] would yield (36, 60, 2918812)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 76 177608 227673 898471 762600 349433 196906 829606 177462 885671 254561 374618 996187 536489 849409 > 27⁶⁰ [i]