MinT - Best Known (140, 140+77, s)-Nets in Base 4

Best Known (140, 140+77, s)-Nets in Base 4

(140, 140+77, 160)-Net over F₄ — Constructive and digital

Digital (140, 217, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 51, 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 (89, 166, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 83, 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]

(140, 140+77, 208)-Net in Base 4 — Constructive

(140, 217, 208)-net in base 4, using

1 times m-reduction [i] based on (140, 218, 208)-net in base 4, using
- trace code for nets [i] based on (31, 109, 104)-net in base 16, using
  - 1 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]

(140, 140+77, 474)-Net over F₄ — Digital

Digital (140, 217, 474)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+77, 13209)-Net in Base 4 — Upper bound on s

There is no (140, 217, 13210)-net in base 4, because

1 times m-reduction [i] would yield (140, 216, 13210)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11098 128865 538394 981555 555589 473479 829191 463501 964896 717123 301401 091562 386239 500774 417824 260950 152329 491501 160242 983377 152300 169040 > 4²¹⁶ [i]