MinT - Best Known (69−64, 69, s)-Nets in Base 8

Best Known (69−64, 69, s)-Nets in Base 8

(69−64, 69, 28)-Net over F₈ — Constructive and digital

Digital (5, 69, 28)-net over F₈, using

net from sequence [i] based on digital (5, 27)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
  - a function field by SÃ©mirat [i]

(69−64, 69, 29)-Net over F₈ — Digital

Digital (5, 69, 29)-net over F₈, using

net from sequence [i] based on digital (5, 28)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(69−64, 69, 49)-Net over F₈ — Upper bound on s (digital)

There is no digital (5, 69, 50)-net over F₈, because

24 times m-reduction [i] would yield digital (5, 45, 50)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8⁴⁵, 50, F₈, 40) (dual of [50, 5, 41]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(8⁴⁴, 46, S₈, 40), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 261336 857795 280739 939871 698507 597986 398208 / 41 > 8⁴⁴ [i]
    2. OA(8⁵, 50, S₈, 4), but
      - discarding factors would yield OA(8⁵, 37, S₈, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 32894 > 8⁵ [i]

(69−64, 69, 51)-Net in Base 8 — Upper bound on s

There is no (5, 69, 52)-net in base 8, because

24 times m-reduction [i] would yield (5, 45, 52)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8⁴⁵, 52, S₈, 40), but
  - the linear programming bound shows that MÂ â‰¥ 17082 370822 074457 326416 360196 848644 913580 998656 / 353625 > 8⁴⁵ [i]