MinT - Best Known (6, 6+40, s)-Nets in Base 8

Best Known (6, 6+40, s)-Nets in Base 8

(6, 6+40, 28)-Net over F₈ — Constructive and digital

Digital (6, 46, 28)-net over F₈, using

t-expansion [i] based on digital (5, 46, 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]

(6, 6+40, 33)-Net over F₈ — Digital

Digital (6, 46, 33)-net over F₈, using

net from sequence [i] based on digital (6, 32)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 6 and N(F) ≥ 33, using
  - Drinfeld modules of rank 1 [i]

(6, 6+40, 76)-Net over F₈ — Upper bound on s (digital)

There is no digital (6, 46, 77)-net over F₈, because

extracting embedded orthogonal array [i] would yield linear OA(8⁴⁶, 77, F₈, 40) (dual of [77, 31, 41]-code), but
- residual code [i] would yield OA(8⁶, 36, S₈, 5), but
  - 1 times truncation [i] would yield OA(8⁵, 35, S₈, 4), but
    - the linear programming bound shows that MÂ â‰¥ 1 306624 / 39 > 8⁵ [i]

(6, 6+40, 86)-Net in Base 8 — Upper bound on s

There is no (6, 46, 87)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8⁴⁶, 87, S₈, 40), but
- the linear programming bound shows that MÂ â‰¥ 975488 403546 292246 689063 457110 274417 970339 192636 558792 824486 038440 405232 075006 803968 / 2 741543 659930 478674 463939 169737 644006 097187 > 8⁴⁶ [i]