MinT - Best Known (142−48, 142, s)-Nets in Base 4

Best Known (142−48, 142, s)-Nets in Base 4

(142−48, 142, 157)-Net over F₄ — Constructive and digital

Digital (94, 142, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 34, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (60, 108, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 54, 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]

(142−48, 142, 196)-Net in Base 4 — Constructive

(94, 142, 196)-net in base 4, using

2 times m-reduction [i] based on (94, 144, 196)-net in base 4, using
- trace code for nets [i] based on (22, 72, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (22, 75, 98)-net in base 16, using
    - base change [i] based on digital (7, 60, 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]

(142−48, 142, 391)-Net over F₄ — Digital

Digital (94, 142, 391)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−48, 142, 11904)-Net in Base 4 — Upper bound on s

There is no (94, 142, 11905)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31 099827 090960 026290 194232 708447 368787 365833 062329 567104 040618 599561 829696 458870 144656 > 4¹⁴² [i]