MinT - Best Known (156, 156+79, s)-Nets in Base 4

Best Known (156, 156+79, s)-Nets in Base 4

(156, 156+79, 170)-Net over F₄ — Constructive and digital

Digital (156, 235, 170)-net over F₄, using

4¹ times duplication [i] based on digital (155, 234, 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, 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]

(156, 156+79, 240)-Net in Base 4 — Constructive

(156, 235, 240)-net in base 4, using

t-expansion [i] based on (155, 235, 240)-net in base 4, using
- 5 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]

(156, 156+79, 615)-Net over F₄ — Digital

Digital (156, 235, 615)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 156+79, 20989)-Net in Base 4 — Upper bound on s

There is no (156, 235, 20990)-net in base 4, because

1 times m-reduction [i] would yield (156, 234, 20990)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 762 809438 800142 990394 157005 521064 179466 735253 063586 166130 120812 329308 544044 528927 888189 068877 275421 694276 686664 954811 165903 316098 416966 162788 > 4²³⁴ [i]