MinT - Best Known (235−81, 235, s)-Nets in Base 4

Best Known (235−81, 235, s)-Nets in Base 4

(235−81, 235, 164)-Net over F₄ — Constructive and digital

Digital (154, 235, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 61, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (93, 174, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 87, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(235−81, 235, 240)-Net in Base 4 — Constructive

(154, 235, 240)-net in base 4, using

3 times m-reduction [i] based on (154, 238, 240)-net in base 4, using
- trace code for nets [i] based on (35, 119, 120)-net in base 16, using
  - 1 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]

(235−81, 235, 562)-Net over F₄ — Digital

Digital (154, 235, 562)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−81, 235, 17454)-Net in Base 4 — Upper bound on s

There is no (154, 235, 17455)-net in base 4, because

1 times m-reduction [i] would yield (154, 234, 17455)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 763 171058 589307 582397 001181 233514 372730 484697 584563 896047 913708 859026 751316 437009 205644 888326 760267 221723 299972 379852 329145 619790 935543 534947 > 4²³⁴ [i]