MinT - Best Known (142, 142+73, s)-Nets in Base 4

Best Known (142, 142+73, s)-Nets in Base 4

(142, 142+73, 164)-Net over F₄ — Constructive and digital

Digital (142, 215, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 57, 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 (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]

(142, 142+73, 240)-Net in Base 4 — Constructive

(142, 215, 240)-net in base 4, using

3 times m-reduction [i] based on (142, 218, 240)-net in base 4, using
- trace code for nets [i] based on (33, 109, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (33, 110, 120)-net in base 16, using
    - base change [i] based on digital (11, 88, 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]

(142, 142+73, 548)-Net over F₄ — Digital

Digital (142, 215, 548)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(142, 142+73, 18022)-Net in Base 4 — Upper bound on s

There is no (142, 215, 18023)-net in base 4, because

1 times m-reduction [i] would yield (142, 214, 18023)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 693 679316 440203 706367 608307 860507 943111 181137 097565 259466 144813 504088 628858 037900 043650 517038 964181 292942 190017 120548 061883 347745 > 4²¹⁴ [i]