MinT - Best Known (135, 203, s)-Nets in Base 4

Best Known (135, 203, s)-Nets in Base 4

(135, 203, 164)-Net over F₄ — Constructive and digital

Digital (135, 203, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 55, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (80, 148, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 74, 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]

(135, 203, 240)-Net in Base 4 — Constructive

(135, 203, 240)-net in base 4, using

3 times m-reduction [i] based on (135, 206, 240)-net in base 4, using
- trace code for nets [i] based on (32, 103, 120)-net in base 16, using
  - 2 times m-reduction [i] based on (32, 105, 120)-net in base 16, using
    - base change [i] based on digital (11, 84, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(135, 203, 547)-Net over F₄ — Digital

Digital (135, 203, 547)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 203, 17714)-Net in Base 4 — Upper bound on s

There is no (135, 203, 17715)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 472260 264170 653886 500580 940824 709873 710038 904447 711914 235433 927982 350091 810691 590039 232040 826063 495980 802958 728400 903095 > 4²⁰³ [i]