MinT - Best Known (131−45, 131, s)-Nets in Base 4

Best Known (131−45, 131, s)-Nets in Base 4

(131−45, 131, 151)-Net over F₄ — Constructive and digital

Digital (86, 131, 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 (57, 102, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 51, 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]

(131−45, 131, 196)-Net in Base 4 — Constructive

(86, 131, 196)-net in base 4, using

4¹ times duplication [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]

(131−45, 131, 345)-Net over F₄ — Digital

Digital (86, 131, 345)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(131−45, 131, 10880)-Net in Base 4 — Upper bound on s

There is no (86, 131, 10881)-net in base 4, because

1 times m-reduction [i] would yield (86, 130, 10881)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 854604 854985 025093 925212 520432 330191 652008 293309 669107 823112 506919 202712 542784 > 4¹³⁰ [i]