MinT - Best Known (172−148, 172, s)-Nets in Base 8

Best Known (172−148, 172, s)-Nets in Base 8

(172−148, 172, 65)-Net over F₈ — Constructive and digital

Digital (24, 172, 65)-net over F₈, using

t-expansion [i] based on digital (14, 172, 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]

(172−148, 172, 81)-Net over F₈ — Digital

Digital (24, 172, 81)-net over F₈, using

net from sequence [i] based on digital (24, 80)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 24 and N(F) ≥ 81, using
  - a curve from the manYPoints database [i]

(172−148, 172, 298)-Net over F₈ — Upper bound on s (digital)

There is no digital (24, 172, 299)-net over F₈, because

4 times m-reduction [i] would yield digital (24, 168, 299)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8¹⁶⁸, 299, F₈, 144) (dual of [299, 131, 145]-code), but
  - residual code [i] would yield OA(8²⁴, 154, S₈, 18), but
    - the linear programming bound shows that MÂ â‰¥ 1463 606729 993253 457700 255083 689454 600192 / 305843 519426 437217 > 8²⁴ [i]

(172−148, 172, 449)-Net in Base 8 — Upper bound on s

There is no (24, 172, 450)-net in base 8, because

28 times m-reduction [i] would yield (24, 144, 450)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11170 651830 433015 653403 055602 867942 032543 601580 266295 219777 697700 093808 528299 244484 622191 586360 733694 092701 701151 245039 147744 656912 > 8¹⁴⁴ [i]