MinT - Best Known (162−53, 162, s)-Nets in Base 4

Best Known (162−53, 162, s)-Nets in Base 4

(162−53, 162, 240)-Net over F₄ — Constructive and digital

Digital (109, 162, 240)-net over F₄, using

trace code for nets [i] based on digital (1, 54, 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]

(162−53, 162, 493)-Net over F₄ — Digital

Digital (109, 162, 493)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−53, 162, 18785)-Net in Base 4 — Upper bound on s

There is no (109, 162, 18786)-net in base 4, because

1 times m-reduction [i] would yield (109, 161, 18786)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 549403 099482 195513 624211 256249 805306 264123 243338 397865 011859 589835 557731 947828 149606 004329 372528 > 4¹⁶¹ [i]