MinT - Best Known (252−92, 252, s)-Nets in Base 4

Best Known (252−92, 252, s)-Nets in Base 4

(252−92, 252, 160)-Net over F₄ — Constructive and digital

Digital (160, 252, 160)-net over F₄, using

t-expansion [i] based on digital (157, 252, 160)-net over F₄, using
- 7 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 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, 175, 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]

(252−92, 252, 196)-Net in Base 4 — Constructive

(160, 252, 196)-net in base 4, using

2 times m-reduction [i] based on (160, 254, 196)-net in base 4, using
- trace code for nets [i] based on (33, 127, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (33, 130, 98)-net in base 16, using
    - base change [i] based on digital (7, 104, 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]

(252−92, 252, 489)-Net over F₄ — Digital

Digital (160, 252, 489)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−92, 252, 11884)-Net in Base 4 — Upper bound on s

There is no (160, 252, 11885)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 527518 890554 286022 943592 976284 734275 220630 131147 069922 029364 460146 987607 651920 594342 525661 357794 366592 800572 454549 101864 329415 749447 028094 170737 474096 > 4²⁵² [i]