MinT - Best Known (145−53, 145, s)-Nets in Base 4

Best Known (145−53, 145, s)-Nets in Base 4

(145−53, 145, 139)-Net over F₄ — Constructive and digital

Digital (92, 145, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 27, 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 (65, 118, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 59, 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]

(145−53, 145, 301)-Net over F₄ — Digital

Digital (92, 145, 301)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(145−53, 145, 7576)-Net in Base 4 — Upper bound on s

There is no (92, 145, 7577)-net in base 4, because

1 times m-reduction [i] would yield (92, 144, 7577)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 498 803048 494038 759114 157064 924874 086232 403462 318849 503159 309832 827920 826241 149585 761296 > 4¹⁴⁴ [i]