MinT - Best Known (175−65, 175, s)-Nets in Base 4

Best Known (175−65, 175, s)-Nets in Base 4

(175−65, 175, 139)-Net over F₄ — Constructive and digital

Digital (110, 175, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 33, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (77, 142, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 71, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(175−65, 175, 336)-Net over F₄ — Digital

Digital (110, 175, 336)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(175−65, 175, 7980)-Net in Base 4 — Upper bound on s

There is no (110, 175, 7981)-net in base 4, because

1 times m-reduction [i] would yield (110, 174, 7981)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 574 083765 868499 128772 404924 393056 472987 322756 914111 600788 139334 777097 709927 900140 427640 339238 142061 766397 > 4¹⁷⁴ [i]