MinT - Best Known (172−59, 172, s)-Nets in Base 4

Best Known (172−59, 172, s)-Nets in Base 4

(172−59, 172, 160)-Net over F₄ — Constructive and digital

Digital (113, 172, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 42, 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 (71, 130, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 65, 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]

(172−59, 172, 208)-Net in Base 4 — Constructive

(113, 172, 208)-net in base 4, using

4² times duplication [i] based on (111, 170, 208)-net in base 4, using
- trace code for nets [i] based on (26, 85, 104)-net in base 16, using
  - base change [i] based on digital (9, 68, 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]

(172−59, 172, 435)-Net over F₄ — Digital

Digital (113, 172, 435)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−59, 172, 13782)-Net in Base 4 — Upper bound on s

There is no (113, 172, 13783)-net in base 4, because

1 times m-reduction [i] would yield (113, 171, 13783)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 964781 325391 015967 776919 951687 793937 882166 719846 664746 907615 043023 758447 755562 393567 919206 783637 542832 > 4¹⁷¹ [i]