MinT - Best Known (207−75, 207, s)-Nets in Base 4

Best Known (207−75, 207, s)-Nets in Base 4

(207−75, 207, 151)-Net over F₄ — Constructive and digital

Digital (132, 207, 151)-net over F₄, using

4¹ times duplication [i] based on digital (131, 206, 151)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 44, 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 (87, 162, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 81, 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]

(207−75, 207, 196)-Net in Base 4 — Constructive

(132, 207, 196)-net in base 4, using

1 times m-reduction [i] based on (132, 208, 196)-net in base 4, using
- trace code for nets [i] based on (28, 104, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (28, 105, 98)-net in base 16, using
    - base change [i] based on digital (7, 84, 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]

(207−75, 207, 423)-Net over F₄ — Digital

Digital (132, 207, 423)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−75, 207, 10954)-Net in Base 4 — Upper bound on s

There is no (132, 207, 10955)-net in base 4, because

1 times m-reduction [i] would yield (132, 206, 10955)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10587 476932 546698 933505 857444 487304 674823 808117 478598 448091 578462 511271 772488 265505 448610 471050 221676 690717 618471 989844 212408 > 4²⁰⁶ [i]