MinT - Best Known (135, 135+74, s)-Nets in Base 4

Best Known (135, 135+74, s)-Nets in Base 4

(135, 135+74, 158)-Net over F₄ — Constructive and digital

Digital (135, 209, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 49, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (86, 160, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 80, 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, 135+74, 208)-Net in Base 4 — Constructive

(135, 209, 208)-net in base 4, using

1 times m-reduction [i] based on (135, 210, 208)-net in base 4, using
- trace code for nets [i] based on (30, 105, 104)-net in base 16, using
  - base change [i] based on digital (9, 84, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(135, 135+74, 461)-Net over F₄ — Digital

Digital (135, 209, 461)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+74, 12261)-Net in Base 4 — Upper bound on s

There is no (135, 209, 12262)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 677925 888127 297741 541415 294833 750157 934290 830117 042362 518704 013084 819655 060092 214027 935217 411279 999724 985247 067362 488446 523445 > 4²⁰⁹ [i]