MinT - Best Known (162−57, 162, s)-Nets in Base 4

Best Known (162−57, 162, s)-Nets in Base 4

(162−57, 162, 151)-Net over F₄ — Constructive and digital

Digital (105, 162, 151)-net over F₄, using

4¹ times duplication [i] based on digital (104, 161, 151)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 35, 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 (69, 126, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 63, 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]

(162−57, 162, 196)-Net in Base 4 — Constructive

(105, 162, 196)-net in base 4, using

4² times duplication [i] based on (103, 160, 196)-net in base 4, using
- trace code for nets [i] based on (23, 80, 98)-net in base 16, using
  - base change [i] based on digital (7, 64, 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]

(162−57, 162, 378)-Net over F₄ — Digital

Digital (105, 162, 378)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−57, 162, 10884)-Net in Base 4 — Upper bound on s

There is no (105, 162, 10885)-net in base 4, because

1 times m-reduction [i] would yield (105, 161, 10885)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 548927 734773 543733 086542 888508 518332 652470 539517 227066 430474 629980 549937 772073 089212 043976 339504 > 4¹⁶¹ [i]