MinT - Best Known (170−57, 170, s)-Nets in Base 4

Best Known (170−57, 170, s)-Nets in Base 4

(170−57, 170, 163)-Net over F₄ — Constructive and digital

Digital (113, 170, 163)-net over F₄, using

4¹ times duplication [i] based on digital (112, 169, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 43, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, 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]

(170−57, 170, 240)-Net in Base 4 — Constructive

(113, 170, 240)-net in base 4, using

trace code for nets [i] based on (28, 85, 120)-net in base 16, using
- base change [i] based on digital (11, 68, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(170−57, 170, 468)-Net over F₄ — Digital

Digital (113, 170, 468)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−57, 170, 16185)-Net in Base 4 — Upper bound on s

There is no (113, 170, 16186)-net in base 4, because

1 times m-reduction [i] would yield (113, 169, 16186)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 560136 943849 071283 609104 642776 233188 462208 274142 838342 255733 655216 507384 424069 888410 850963 198506 527320 > 4¹⁶⁹ [i]