MinT - Best Known (91, 139, s)-Nets in Base 4

Best Known (91, 139, s)-Nets in Base 4

(91, 139, 151)-Net over F₄ — Constructive and digital

Digital (91, 139, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 31, 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 (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]

(91, 139, 196)-Net in Base 4 — Constructive

(91, 139, 196)-net in base 4, using

1 times m-reduction [i] based on (91, 140, 196)-net in base 4, using
- trace code for nets [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]

(91, 139, 355)-Net over F₄ — Digital

Digital (91, 139, 355)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(91, 139, 10007)-Net in Base 4 — Upper bound on s

There is no (91, 139, 10008)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 486198 209464 701041 391412 169046 352738 388882 626199 562262 052810 294668 312960 292561 878144 > 4¹³⁹ [i]