MinT - Best Known (136−47, 136, s)-Nets in Base 4

Best Known (136−47, 136, s)-Nets in Base 4

(136−47, 136, 151)-Net over F₄ — Constructive and digital

Digital (89, 136, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 30, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (59, 106, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 53, 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]

(136−47, 136, 196)-Net in Base 4 — Constructive

(89, 136, 196)-net in base 4, using

trace code for nets [i] based on (21, 68, 98)-net in base 16, using
- 2 times m-reduction [i] based on (21, 70, 98)-net in base 16, using
  - base change [i] based on digital (7, 56, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(136−47, 136, 348)-Net over F₄ — Digital

Digital (89, 136, 348)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−47, 136, 10725)-Net in Base 4 — Upper bound on s

There is no (89, 136, 10726)-net in base 4, because

1 times m-reduction [i] would yield (89, 135, 10726)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1899 855567 694164 921525 443405 695468 847769 082294 538290 632856 974192 006936 209606 332144 > 4¹³⁵ [i]