MinT - Best Known (56−53, 56, s)-Nets in Base 8

Best Known (56−53, 56, s)-Nets in Base 8

(56−53, 56, 24)-Net over F₈ — Constructive and digital

Digital (3, 56, 24)-net over F₈, using

net from sequence [i] based on digital (3, 23)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
  - the Klein quartic over F₈ [i]

(56−53, 56, 33)-Net over F₈ — Upper bound on s (digital)

There is no digital (3, 56, 34)-net over F₈, because

29 times m-reduction [i] would yield digital (3, 27, 34)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8²⁷, 34, F₈, 24) (dual of [34, 7, 25]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(8²⁶, 28, S₈, 24), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 9 671406 556917 033397 649408 / 25 > 8²⁶ [i]
    2. OA(8⁷, 34, S₈, 6), but
      - discarding factors would yield OA(8⁷, 32, S₈, 6), but
        the linear programming bound shows that MÂ â‰¥ 3784 900608 / 1729 > 8⁷ [i]

(56−53, 56, 36)-Net in Base 8 — Upper bound on s

There is no (3, 56, 37)-net in base 8, because

23 times m-reduction [i] would yield (3, 33, 37)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8³³, 37, S₈, 30), but
  - the linear programming bound shows that MÂ â‰¥ 299 165541 653862 138753 221956 468736 / 465 > 8³³ [i]