MinT - Best Known (138, 138+75, s)-Nets in Base 4

Best Known (138, 138+75, s)-Nets in Base 4

(138, 138+75, 160)-Net over F₄ — Constructive and digital

Digital (138, 213, 160)-net over F₄, using

4¹ times duplication [i] based on digital (137, 212, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 50, 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 (87, 162, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 81, 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]

(138, 138+75, 208)-Net in Base 4 — Constructive

(138, 213, 208)-net in base 4, using

1 times m-reduction [i] based on (138, 214, 208)-net in base 4, using
- trace code for nets [i] based on (31, 107, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (31, 110, 104)-net in base 16, using
    - base change [i] based on digital (9, 88, 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]

(138, 138+75, 478)-Net over F₄ — Digital

Digital (138, 213, 478)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+75, 13723)-Net in Base 4 — Upper bound on s

There is no (138, 213, 13724)-net in base 4, because

1 times m-reduction [i] would yield (138, 212, 13724)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 349368 438589 250217 046937 249749 430146 817562 999712 551883 270811 392000 934141 550384 778611 297353 634144 873622 874068 323201 327339 980520 > 4²¹² [i]