MinT - Best Known (159, 159+85, s)-Nets in Base 4

Best Known (159, 159+85, s)-Nets in Base 4

(159, 159+85, 163)-Net over F₄ — Constructive and digital

Digital (159, 244, 163)-net over F₄, using

3 times m-reduction [i] based on digital (159, 247, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 59, 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 (100, 188, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 94, 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]

(159, 159+85, 240)-Net in Base 4 — Constructive

(159, 244, 240)-net in base 4, using

2 times m-reduction [i] based on (159, 246, 240)-net in base 4, using
- trace code for nets [i] based on (36, 123, 120)-net in base 16, using
  - 2 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 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]

(159, 159+85, 561)-Net over F₄ — Digital

Digital (159, 244, 561)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+85, 16716)-Net in Base 4 — Upper bound on s

There is no (159, 244, 16717)-net in base 4, because

1 times m-reduction [i] would yield (159, 243, 16717)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200 281270 237746 370001 406725 671985 394577 856969 352943 459386 180244 280083 669692 178824 970800 698225 986490 781878 949813 457141 922661 521549 641030 359387 553420 > 4²⁴³ [i]