MinT - Best Known (9, ∞, s)-Nets in Base 8

Best Known (9, ∞, s)-Nets in Base 8

(9, ∞, 45)-Net over F₈ — Constructive and digital

Digital (9, m, 45)-net over F₈ for arbitrarily large m, 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]

(9, ∞, 79)-Net over F₈ — Upper bound on s (digital)

There is no digital (9, m, 80)-net over F₈ for arbitrarily large m, because

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]

(9, ∞, 80)-Net in Base 8 — Upper bound on s

There is no (9, m, 81)-net in base 8 for arbitrarily large m, because

m-reduction [i] would yield (9, 159, 81)-net in base 8, but
- extracting embedded OOA [i] would yield OOA(8¹⁵⁹, 81, S₈, 2, 150), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 78 043713 757899 805784 539930 744829 157643 714953 566624 278771 478923 990634 293470 494140 503007 652576 587299 278995 673278 035165 572386 199391 982207 132657 254400 / 151 > 8¹⁵⁹ [i]