MinT - Best Known (19, 168, s)-Nets in Base 8

Best Known (19, 168, s)-Nets in Base 8

(19, 168, 65)-Net over F₈ — Constructive and digital

Digital (19, 168, 65)-net over F₈, using

t-expansion [i] based on digital (14, 168, 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]

(19, 168, 160)-Net over F₈ — Upper bound on s (digital)

There is no digital (19, 168, 161)-net over F₈, because

13 times m-reduction [i] would yield digital (19, 155, 161)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8¹⁵⁵, 161, F₈, 136) (dual of [161, 6, 137]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(8¹⁵⁴, 157, F₈, 136) (dual of [157, 3, 137]-code), but
      - Griesmer bound [i]
    2. OA(8⁶, 161, S₈, 4), but
      - discarding factors would yield OA(8⁶, 104, S₈, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 263173 > 8⁶ [i]

(19, 168, 170)-Net in Base 8 — Upper bound on s

There is no (19, 168, 171)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8¹⁶⁸, 171, S₈, 149), but
- the (dual) Plotkin bound shows that MÂ â‰¥ 1675 975991 242824 637446 753124 775730 765934 920727 574049 172215 445180 465220 503759 193372 100234 287270 862928 461253 982273 310756 356719 235351 493321 243304 206125 760512 / 25 > 8¹⁶⁸ [i]