MinT - Best Known (134−46, 134, s)-Nets in Base 4

Best Known (134−46, 134, s)-Nets in Base 4

(134−46, 134, 151)-Net over F₄ — Constructive and digital

Digital (88, 134, 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 (58, 104, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 52, 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]

(134−46, 134, 196)-Net in Base 4 — Constructive

(88, 134, 196)-net in base 4, using

trace code for nets [i] based on (21, 67, 98)-net in base 16, using
- 3 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]

(134−46, 134, 352)-Net over F₄ — Digital

Digital (88, 134, 352)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(134−46, 134, 10096)-Net in Base 4 — Upper bound on s

There is no (88, 134, 10097)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 474 411324 860162 332999 059156 303023 643581 595774 454759 197333 413474 392313 787703 399648 > 4¹³⁴ [i]