MinT - Best Known (153, 230, s)-Nets in Base 4

Best Known (153, 230, s)-Nets in Base 4

(153, 230, 170)-Net over F₄ — Constructive and digital

Digital (153, 230, 170)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (42, 80, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 40, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (73, 150, 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]

(153, 230, 240)-Net in Base 4 — Constructive

(153, 230, 240)-net in base 4, using

6 times m-reduction [i] based on (153, 236, 240)-net in base 4, using
- trace code for nets [i] based on (35, 118, 120)-net in base 16, using
  - 2 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
    - base change [i] based on digital (11, 96, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(153, 230, 612)-Net over F₄ — Digital

Digital (153, 230, 612)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(153, 230, 21244)-Net in Base 4 — Upper bound on s

There is no (153, 230, 21245)-net in base 4, because

1 times m-reduction [i] would yield (153, 229, 21245)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 744770 074843 593305 043259 685382 132858 341681 338414 345598 588386 573108 171581 907616 760197 106208 254078 263917 697366 262103 563937 742327 183874 694900 > 4²²⁹ [i]