MinT - Best Known (137−128, 137, s)-Nets in Base 8

Best Known (137−128, 137, s)-Nets in Base 8

(137−128, 137, 45)-Net over F₈ — Constructive and digital

Digital (9, 137, 45)-net over F₈, using

net from sequence [i] based on digital (9, 44)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
  - an explicitly constructive algebraic function field [i]

(137−128, 137, 79)-Net over F₈ — Upper bound on s (digital)

There is no digital (9, 137, 80)-net over F₈, because

64 times m-reduction [i] would yield digital (9, 73, 80)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8⁷³, 80, F₈, 64) (dual of [80, 7, 65]-code), but
  - residual code [i] would yield linear OA(8⁹, 15, F₈, 8) (dual of [15, 6, 9]-code), but
    - â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]

(137−128, 137, 81)-Net in Base 8 — Upper bound on s

There is no (9, 137, 82)-net in base 8, because

65 times m-reduction [i] would yield (9, 72, 82)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8⁷², 82, S₈, 63), but
  - the linear programming bound shows that MÂ â‰¥ 5 943562 882293 362898 492495 479069 260758 679371 222248 872068 611620 811931 910144 / 44 417945 > 8⁷² [i]