MinT - Best Known (111, 172, s)-Nets in Base 4

Best Known (111, 172, s)-Nets in Base 4

(111, 172, 151)-Net over F₄ — Constructive and digital

Digital (111, 172, 151)-net over F₄, using

4¹ times duplication [i] based on digital (110, 171, 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 (73, 134, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 67, 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]

(111, 172, 196)-Net in Base 4 — Constructive

(111, 172, 196)-net in base 4, using

4² times duplication [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]

(111, 172, 387)-Net over F₄ — Digital

Digital (111, 172, 387)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 172, 10825)-Net in Base 4 — Upper bound on s

There is no (111, 172, 10826)-net in base 4, because

1 times m-reduction [i] would yield (111, 171, 10826)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 983479 802220 376970 172328 430899 864602 640683 083257 633558 484155 405072 048709 410089 012415 303212 719243 234912 > 4¹⁷¹ [i]