MinT - Best Known (162−85, 162, s)-Nets in Base 8

Best Known (162−85, 162, s)-Nets in Base 8

(162−85, 162, 130)-Net over F₈ — Constructive and digital

Digital (77, 162, 130)-net over F₈, using

t-expansion [i] based on digital (76, 162, 130)-net over F₈, using
- 10 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
    2. digital (14, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(162−85, 162, 215)-Net over F₈ — Digital

Digital (77, 162, 215)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−85, 162, 6805)-Net in Base 8 — Upper bound on s

There is no (77, 162, 6806)-net in base 8, because

1 times m-reduction [i] would yield (77, 161, 6806)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 043837 089058 409466 540998 530843 282788 985213 374933 194694 199543 966161 619450 348989 259310 141525 182095 171308 505122 884420 424936 002653 858158 838324 067728 > 8¹⁶¹ [i]