MinT - Best Known (138, 138+70, s)-Nets in Base 4

Best Known (138, 138+70, s)-Nets in Base 4

(138, 138+70, 164)-Net over F₄ — Constructive and digital

Digital (138, 208, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 56, 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 (82, 152, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 76, 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+70, 240)-Net in Base 4 — Constructive

(138, 208, 240)-net in base 4, using

t-expansion [i] based on (137, 208, 240)-net in base 4, using
- 2 times m-reduction [i] based on (137, 210, 240)-net in base 4, using
  - trace code for nets [i] based on (32, 105, 120)-net in base 16, using
    - base change [i] based on digital (11, 84, 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]

(138, 138+70, 550)-Net over F₄ — Digital

Digital (138, 208, 550)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+70, 17514)-Net in Base 4 — Upper bound on s

There is no (138, 208, 17515)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169515 161910 551762 510468 956560 257143 235000 715403 203610 996531 671916 118903 110632 794160 775874 001382 227590 653665 113587 201145 853772 > 4²⁰⁸ [i]