MinT - Best Known (238−81, 238, s)-Nets in Base 4

Best Known (238−81, 238, s)-Nets in Base 4

(238−81, 238, 170)-Net over F₄ — Constructive and digital

Digital (157, 238, 170)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (44, 84, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 42, 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, 154, 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]

(238−81, 238, 240)-Net in Base 4 — Constructive

(157, 238, 240)-net in base 4, using

t-expansion [i] based on (155, 238, 240)-net in base 4, using
- 2 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]

(238−81, 238, 595)-Net over F₄ — Digital

Digital (157, 238, 595)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−81, 238, 19370)-Net in Base 4 — Upper bound on s

There is no (157, 238, 19371)-net in base 4, because

1 times m-reduction [i] would yield (157, 237, 19371)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48830 086684 755290 250606 896236 897559 558883 584123 713740 093865 927054 365565 768320 201319 514894 292707 984978 035487 962740 158076 143303 208161 843472 254641 > 4²³⁷ [i]