MinT - Best Known (208−73, 208, s)-Nets in Base 4

Best Known (208−73, 208, s)-Nets in Base 4

(208−73, 208, 160)-Net over F₄ — Constructive and digital

Digital (135, 208, 160)-net over F₄, using

4¹ times duplication [i] based on digital (134, 207, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 49, 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 (85, 158, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 79, 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]

(208−73, 208, 208)-Net in Base 4 — Constructive

(135, 208, 208)-net in base 4, using

2 times m-reduction [i] based on (135, 210, 208)-net in base 4, using
- trace code for nets [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]

(208−73, 208, 474)-Net over F₄ — Digital

Digital (135, 208, 474)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(208−73, 208, 13757)-Net in Base 4 — Upper bound on s

There is no (135, 208, 13758)-net in base 4, because

1 times m-reduction [i] would yield (135, 207, 13758)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42391 765249 225657 472523 298349 355590 215569 352572 346026 173085 366874 702037 840402 516568 880617 931275 399992 362733 569314 559632 487105 > 4²⁰⁷ [i]