MinT - Best Known (27, 172, s)-Nets in Base 8

Best Known (27, 172, s)-Nets in Base 8

(27, 172, 65)-Net over F₈ — Constructive and digital

Digital (27, 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]

(27, 172, 96)-Net over F₈ — Digital

Digital (27, 172, 96)-net over F₈, using

net from sequence [i] based on digital (27, 95)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 27 and N(F) ≥ 96, using
  - a curve from the manYPoints database [i]

(27, 172, 452)-Net over F₈ — Upper bound on s (digital)

There is no digital (27, 172, 453)-net over F₈, because

1 times m-reduction [i] would yield digital (27, 171, 453)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8¹⁷¹, 453, F₈, 144) (dual of [453, 282, 145]-code), but
  - residual code [i] would yield OA(8²⁷, 308, S₈, 18), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 2 475739 441552 887337 471106 > 8²⁷ [i]

(27, 172, 502)-Net in Base 8 — Upper bound on s

There is no (27, 172, 503)-net in base 8, because

15 times m-reduction [i] would yield (27, 157, 503)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6252 405981 624735 318231 904139 143556 913561 824651 858378 143178 859892 009294 431150 085144 868249 581870 432507 375686 473585 361068 102909 505869 323664 914818 > 8¹⁵⁷ [i]