MinT - Best Known (172−55, 172, s)-Nets in Base 4

Best Known (172−55, 172, s)-Nets in Base 4

(172−55, 172, 240)-Net over F₄ — Constructive and digital

Digital (117, 172, 240)-net over F₄, using

2 times m-reduction [i] based on digital (117, 174, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 58, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(172−55, 172, 566)-Net over F₄ — Digital

Digital (117, 172, 566)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−55, 172, 23655)-Net in Base 4 — Upper bound on s

There is no (117, 172, 23656)-net in base 4, because

1 times m-reduction [i] would yield (117, 171, 23656)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 966080 407340 588247 755749 722819 450753 589176 297670 116589 322654 746898 645983 567779 109989 948730 826393 874332 > 4¹⁷¹ [i]