MinT - Best Known (127, 127+68, s)-Nets in Base 4

Best Known (127, 127+68, s)-Nets in Base 4

(127, 127+68, 160)-Net over F₄ — Constructive and digital

Digital (127, 195, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 47, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the 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]

(127, 127+68, 208)-Net in Base 4 — Constructive

(127, 195, 208)-net in base 4, using

1 times m-reduction [i] based on (127, 196, 208)-net in base 4, using
- trace code for nets [i] based on (29, 98, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (29, 100, 104)-net in base 16, using
    - base change [i] based on digital (9, 80, 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]

(127, 127+68, 458)-Net over F₄ — Digital

Digital (127, 195, 458)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+68, 12776)-Net in Base 4 — Upper bound on s

There is no (127, 195, 12777)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2527 230899 123899 020449 268885 260198 322905 772163 408276 840521 169205 462266 792108 114010 356473 563025 472394 291925 939808 428532 > 4¹⁹⁵ [i]