MinT - Best Known (162−55, 162, s)-Nets in Base 4

Best Known (162−55, 162, s)-Nets in Base 4

(162−55, 162, 160)-Net over F₄ — Constructive and digital

Digital (107, 162, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 40, 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 (67, 122, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 61, 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]

(162−55, 162, 208)-Net in Base 4 — Constructive

(107, 162, 208)-net in base 4, using

4² times duplication [i] based on (105, 160, 208)-net in base 4, using
- trace code for nets [i] based on (25, 80, 104)-net in base 16, using
  - base change [i] based on digital (9, 64, 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]

(162−55, 162, 429)-Net over F₄ — Digital

Digital (107, 162, 429)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−55, 162, 14147)-Net in Base 4 — Upper bound on s

There is no (107, 162, 14148)-net in base 4, because

1 times m-reduction [i] would yield (107, 161, 14148)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 555909 265832 001703 753363 061378 791837 608694 162228 996719 222625 016232 034826 460118 362337 333731 177272 > 4¹⁶¹ [i]