MinT - Best Known (233−78, 233, s)-Nets in Base 4

Best Known (233−78, 233, s)-Nets in Base 4

(233−78, 233, 170)-Net over F₄ — Constructive and digital

Digital (155, 233, 170)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (43, 82, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 41, 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, 151, 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]

(233−78, 233, 240)-Net in Base 4 — Constructive

(155, 233, 240)-net in base 4, using

7 times m-reduction [i] based on (155, 240, 240)-net in base 4, using
- trace code for nets [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]

(233−78, 233, 619)-Net over F₄ — Digital

Digital (155, 233, 619)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−78, 233, 20255)-Net in Base 4 — Upper bound on s

There is no (155, 233, 20256)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 190 747091 479099 340470 171285 345192 012759 853116 255883 913147 273398 119518 435365 718481 060326 390338 321682 979371 610134 386176 440314 797685 000625 476920 > 4²³³ [i]