MinT - Best Known (133, 133+80, s)-Nets in Base 4

Best Known (133, 133+80, s)-Nets in Base 4

(133, 133+80, 139)-Net over F₄ — Constructive and digital

Digital (133, 213, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 41, 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 (92, 172, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 86, 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]

(133, 133+80, 382)-Net over F₄ — Digital

Digital (133, 213, 382)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(133, 133+80, 8412)-Net in Base 4 — Upper bound on s

There is no (133, 213, 8413)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 173 302552 286490 805060 880107 966042 262773 305079 945340 004100 043687 354584 125058 424867 708897 781008 849510 154596 631573 676137 790634 233870 > 4²¹³ [i]