MinT - Best Known (136, 136+75, s)-Nets in Base 4

Best Known (136, 136+75, s)-Nets in Base 4

(136, 136+75, 158)-Net over F₄ — Constructive and digital

Digital (136, 211, 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 (87, 162, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 81, 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]

(136, 136+75, 208)-Net in Base 4 — Constructive

(136, 211, 208)-net in base 4, using

4¹ times duplication [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]

(136, 136+75, 459)-Net over F₄ — Digital

Digital (136, 211, 459)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+75, 12730)-Net in Base 4 — Upper bound on s

There is no (136, 211, 12731)-net in base 4, because

1 times m-reduction [i] would yield (136, 210, 12731)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 709372 506840 518975 428935 538921 289049 900641 517372 675315 208482 607342 322115 050898 919934 641918 341241 348788 258367 999470 089628 070700 > 4²¹⁰ [i]