MinT - Best Known (109, 109+60, s)-Nets in Base 4

Best Known (109, 109+60, s)-Nets in Base 4

(109, 109+60, 151)-Net over F₄ — Constructive and digital

Digital (109, 169, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 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 (72, 132, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 66, 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]

(109, 109+60, 196)-Net in Base 4 — Constructive

(109, 169, 196)-net in base 4, using

1 times m-reduction [i] based on (109, 170, 196)-net in base 4, using
- trace code for nets [i] based on (24, 85, 98)-net in base 16, using
  - base change [i] based on digital (7, 68, 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]

(109, 109+60, 379)-Net over F₄ — Digital

Digital (109, 169, 379)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+60, 9867)-Net in Base 4 — Upper bound on s

There is no (109, 169, 9868)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 561214 196649 926368 360156 371744 639997 785847 225059 071539 089056 724750 277834 589571 940555 403841 397836 602848 > 4¹⁶⁹ [i]