MinT - Best Known (192, 192+68, s)-Nets in Base 4

Best Known (192, 192+68, s)-Nets in Base 4

(192, 192+68, 540)-Net over F₄ — Constructive and digital

Digital (192, 260, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 35, 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 (157, 225, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 75, 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]

(192, 192+68, 648)-Net in Base 4 — Constructive

(192, 260, 648)-net in base 4, using

4² times duplication [i] based on (190, 258, 648)-net in base 4, using
- trace code for nets [i] based on (18, 86, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
    - base change [i] based on digital (5, 78, 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]

(192, 192+68, 1898)-Net over F₄ — Digital

Digital (192, 260, 1898)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(192, 192+68, 181249)-Net in Base 4 — Upper bound on s

There is no (192, 260, 181250)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 433031 460800 131529 159046 823610 900133 963620 834339 206922 700357 899311 979089 464741 219884 372473 424825 626056 303787 664242 491378 847792 658441 631963 458066 022883 077376 > 4²⁶⁰ [i]