MinT - Best Known (148, 148+84, s)-Nets in Base 4

Best Known (148, 148+84, s)-Nets in Base 4

(148, 148+84, 160)-Net over F₄ — Constructive and digital

Digital (148, 232, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 75, 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, 157, 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]

(148, 148+84, 196)-Net in Base 4 — Constructive

(148, 232, 196)-net in base 4, using

2 times m-reduction [i] based on (148, 234, 196)-net in base 4, using
- trace code for nets [i] based on (31, 117, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (31, 120, 98)-net in base 16, using
    - base change [i] based on digital (7, 96, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(148, 148+84, 468)-Net over F₄ — Digital

Digital (148, 232, 468)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(148, 148+84, 11616)-Net in Base 4 — Upper bound on s

There is no (148, 232, 11617)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 795890 200990 771474 358313 642205 333859 153943 545662 457334 049165 288299 932231 735266 633104 362698 467069 342884 014320 047274 391612 459134 635453 956640 > 4²³² [i]