MinT - Best Known (168, 168+58, s)-Nets in Base 4

Best Known (168, 168+58, s)-Nets in Base 4

(168, 168+58, 541)-Net over F₄ — Constructive and digital

Digital (168, 226, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 31, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (137, 195, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 65, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(168, 168+58, 648)-Net in Base 4 — Constructive

(168, 226, 648)-net in base 4, using

2 times m-reduction [i] based on (168, 228, 648)-net in base 4, using
- trace code for nets [i] based on (16, 76, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(168, 168+58, 1823)-Net over F₄ — Digital

Digital (168, 226, 1823)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+58, 191365)-Net in Base 4 — Upper bound on s

There is no (168, 226, 191366)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11630 907130 277186 136762 228897 640368 492860 544171 055307 392886 346888 048492 079773 813398 085082 493630 680976 933965 866935 691038 475278 027859 418096 > 4²²⁶ [i]