MinT - Best Known (236−79, 236, s)-Nets in Base 4

Best Known (236−79, 236, s)-Nets in Base 4

(236−79, 236, 180)-Net over F₄ — Constructive and digital

Digital (157, 236, 180)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (45, 84, 76)-net over F₄, using
  - trace code for nets [i] based on digital (3, 42, 38)-net over F₁₆, using
    - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (73, 152, 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]

(236−79, 236, 240)-Net in Base 4 — Constructive

(157, 236, 240)-net in base 4, using

t-expansion [i] based on (155, 236, 240)-net in base 4, using
- 4 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]

(236−79, 236, 626)-Net over F₄ — Digital

Digital (157, 236, 626)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(236−79, 236, 21750)-Net in Base 4 — Upper bound on s

There is no (157, 236, 21751)-net in base 4, because

1 times m-reduction [i] would yield (157, 235, 21751)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3052 950898 441961 253802 416038 315005 157527 372539 372896 613621 055383 648804 587837 183540 506025 883388 052550 455445 884975 406615 448746 903410 031350 783404 > 4²³⁵ [i]