MinT - Best Known (135, 135+32, s)-Nets in Base 4

Best Known (135, 135+32, s)-Nets in Base 4

(135, 135+32, 1104)-Net over F₄ — Constructive and digital

Digital (135, 167, 1104)-net over F₄, using

4¹ times duplication [i] based on digital (134, 166, 1104)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (22, 38, 76)-net over F₄, using
    - trace code for nets [i] based on digital (3, 19, 38)-net over F₁₆, using
      - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
  2. digital (96, 128, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 32, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(135, 135+32, 7264)-Net over F₄ — Digital

Digital (135, 167, 7264)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+32, 4359305)-Net in Base 4 — Upper bound on s

There is no (135, 167, 4359306)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34996 017240 066321 548107 222846 887882 933591 559164 530553 497548 552909 103613 108265 167540 544990 659210 428392 > 4¹⁶⁷ [i]