MinT - Best Known (147, 147+83, s)-Nets in Base 4

Best Known (147, 147+83, s)-Nets in Base 4

(147, 147+83, 160)-Net over F₄ — Constructive and digital

Digital (147, 230, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 74, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 156, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(147, 147+83, 208)-Net in Base 4 — Constructive

(147, 230, 208)-net in base 4, using

trace code for nets [i] based on (32, 115, 104)-net in base 16, using
- base change [i] based on digital (9, 92, 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]

(147, 147+83, 470)-Net over F₄ — Digital

Digital (147, 230, 470)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(147, 147+83, 12369)-Net in Base 4 — Upper bound on s

There is no (147, 230, 12370)-net in base 4, because

1 times m-reduction [i] would yield (147, 229, 12370)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 746527 587231 954444 019104 515627 091008 464081 286409 027790 537655 050980 368141 948119 260915 869556 129171 863221 054638 615578 587906 738975 118400 105292 > 4²²⁹ [i]