MinT - Best Known (105−49, 105, s)-Nets in Base 3

Best Known (105−49, 105, s)-Nets in Base 3

(105−49, 105, 56)-Net over F₃ — Constructive and digital

Digital (56, 105, 56)-net over F₃, using

1 times m-reduction [i] based on digital (56, 106, 56)-net over F₃, using
- trace code for nets [i] based on digital (3, 53, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]

(105−49, 105, 71)-Net over F₃ — Digital

Digital (56, 105, 71)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−49, 105, 549)-Net in Base 3 — Upper bound on s

There is no (56, 105, 550)-net in base 3, because

1 times m-reduction [i] would yield (56, 104, 550)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 42 623725 131464 614207 780151 569547 289731 213910 420081 > 3¹⁰⁴ [i]