MinT - Best Known (129−44, 129, s)-Nets in Base 4

Best Known (129−44, 129, s)-Nets in Base 4

(129−44, 129, 151)-Net over F₄ — Constructive and digital

Digital (85, 129, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 29, 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 (56, 100, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 50, 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]

(129−44, 129, 196)-Net in Base 4 — Constructive

(85, 129, 196)-net in base 4, using

1 times m-reduction [i] based on (85, 130, 196)-net in base 4, using
- trace code for nets [i] based on (20, 65, 98)-net in base 16, using
  - base change [i] based on digital (7, 52, 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]

(129−44, 129, 349)-Net over F₄ — Digital

Digital (85, 129, 349)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−44, 129, 10214)-Net in Base 4 — Upper bound on s

There is no (85, 129, 10215)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 463233 150420 458307 483304 190438 266109 997458 908523 698502 225796 753195 286381 736386 > 4¹²⁹ [i]