MinT - Best Known (4, 4+40, s)-Nets in Base 8

Best Known (4, 4+40, s)-Nets in Base 8

(4, 4+40, 25)-Net over F₈ — Constructive and digital

Digital (4, 44, 25)-net over F₈, using

net from sequence [i] based on digital (4, 24)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
  - a function field by SÃ©mirat [i]

(4, 4+40, 40)-Net over F₈ — Upper bound on s (digital)

There is no digital (4, 44, 41)-net over F₈, because

8 times m-reduction [i] would yield digital (4, 36, 41)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8³⁶, 41, F₈, 32) (dual of [41, 5, 33]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(8³⁵, 37, S₈, 32), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 1622 592768 292133 633915 780102 881280 / 33 > 8³⁵ [i]
    2. OA(8⁵, 41, 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]

(4, 4+40, 44)-Net in Base 8 — Upper bound on s

There is no (4, 44, 45)-net in base 8, because

6 times m-reduction [i] would yield (4, 38, 45)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8³⁸, 45, S₈, 34), but
  - the linear programming bound shows that MÂ â‰¥ 67 126013 787138 251581 642256 544157 401088 / 2795 > 8³⁸ [i]