MinT - Best Known (203−71, 203, s)-Nets in Base 4

Best Known (203−71, 203, s)-Nets in Base 4

(203−71, 203, 160)-Net over F₄ — Constructive and digital

Digital (132, 203, 160)-net over F₄, using

4¹ times duplication [i] based on digital (131, 202, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 48, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (83, 154, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 77, 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]

(203−71, 203, 208)-Net in Base 4 — Constructive

(132, 203, 208)-net in base 4, using

1 times m-reduction [i] based on (132, 204, 208)-net in base 4, using
- trace code for nets [i] based on (30, 102, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (30, 105, 104)-net in base 16, using
    - base change [i] based on digital (9, 84, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(203−71, 203, 469)-Net over F₄ — Digital

Digital (132, 203, 469)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(203−71, 203, 13803)-Net in Base 4 — Upper bound on s

There is no (132, 203, 13804)-net in base 4, because

1 times m-reduction [i] would yield (132, 202, 13804)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 41 370726 049164 656567 190650 501502 052784 318158 559790 136809 687596 176385 339802 220008 935691 021170 269619 651240 635359 700728 991645 > 4²⁰² [i]