MinT - Best Known (6, 6+157, s)-Nets in Base 8

Best Known (6, 6+157, s)-Nets in Base 8

(6, 6+157, 28)-Net over F₈ — Constructive and digital

Digital (6, 163, 28)-net over F₈, using

t-expansion [i] based on digital (5, 163, 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+157, 33)-Net over F₈ — Digital

Digital (6, 163, 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+157, 57)-Net over F₈ — Upper bound on s (digital)

There is no digital (6, 163, 58)-net over F₈, because

115 times m-reduction [i] would yield digital (6, 48, 58)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8⁴⁸, 58, F₈, 42) (dual of [58, 10, 43]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(8⁴⁷, 50, F₈, 42) (dual of [50, 3, 43]-code), but
      - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
    2. OA(8¹⁰, 58, S₈, 8), but
      - discarding factors would yield OA(8¹⁰, 57, S₈, 8), but
        the linear programming bound shows that MÂ â‰¥ 1426 626655 027200 / 1 315507 > 8¹⁰ [i]

(6, 6+157, 58)-Net in Base 8 — Upper bound on s

There is no (6, 163, 59)-net in base 8, because

48 times m-reduction [i] would yield (6, 115, 59)-net in base 8, but
- extracting embedded OOA [i] would yield OOA(8¹¹⁵, 59, S₈, 2, 109), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 5447 059212 976419 840075 699626 342456 484391 841849 274667 668876 773493 543941 830710 151419 166294 063636 548505 567232 / 55 > 8¹¹⁵ [i]