MinT - Best Known (193−67, 193, s)-Nets in Base 4

Best Known (193−67, 193, s)-Nets in Base 4

(193−67, 193, 160)-Net over F₄ — Constructive and digital

Digital (126, 193, 160)-net over F₄, using

4¹ times duplication [i] based on digital (125, 192, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 46, 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 (79, 146, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 73, 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]

(193−67, 193, 208)-Net in Base 4 — Constructive

(126, 193, 208)-net in base 4, using

1 times m-reduction [i] based on (126, 194, 208)-net in base 4, using
- trace code for nets [i] based on (29, 97, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (29, 100, 104)-net in base 16, using
    - base change [i] based on digital (9, 80, 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]

(193−67, 193, 461)-Net over F₄ — Digital

Digital (126, 193, 461)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−67, 193, 13940)-Net in Base 4 — Upper bound on s

There is no (126, 193, 13941)-net in base 4, because

1 times m-reduction [i] would yield (126, 192, 13941)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 39 420728 160322 121134 730251 542378 759976 663910 125828 577592 294386 121648 703813 634016 379830 312619 916836 258246 613198 279680 > 4¹⁹² [i]