MinT - Best Known (245−87, 245, s)-Nets in Base 4

Best Known (245−87, 245, s)-Nets in Base 4

(245−87, 245, 163)-Net over F₄ — Constructive and digital

Digital (158, 245, 163)-net over F₄, using

4¹ times duplication [i] based on digital (157, 244, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 58, 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 (99, 186, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 93, 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]

(245−87, 245, 208)-Net in Base 4 — Constructive

(158, 245, 208)-net in base 4, using

3 times m-reduction [i] based on (158, 248, 208)-net in base 4, using
- trace code for nets [i] based on (34, 124, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (34, 125, 104)-net in base 16, using
    - base change [i] based on digital (9, 100, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(245−87, 245, 526)-Net over F₄ — Digital

Digital (158, 245, 526)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−87, 245, 14643)-Net in Base 4 — Upper bound on s

There is no (158, 245, 14644)-net in base 4, because

1 times m-reduction [i] would yield (158, 244, 14644)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 800 567904 303440 167350 018510 982105 412271 362044 515445 700304 811010 845345 354600 916245 518672 069443 703758 423593 103853 052345 152024 166539 275094 296879 921446 > 4²⁴⁴ [i]