MinT - Best Known (89−80, 89, s)-Nets in Base 7

Best Known (89−80, 89, s)-Nets in Base 7

(89−80, 89, 17)-Net over F₇ — Constructive and digital

Digital (9, 89, 17)-net over F₇, using

net from sequence [i] based on digital (9, 16)-sequence over F₇, using
- Niederreiter sequence [i]

(89−80, 89, 38)-Net over F₇ — Digital

Digital (9, 89, 38)-net over F₇, using

net from sequence [i] based on digital (9, 37)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 9 and N(F) ≥ 38, using
  - an algebraic function field over â„¤₇ by Yeo [i]

(89−80, 89, 70)-Net over F₇ — Upper bound on s (digital)

There is no digital (9, 89, 71)-net over F₇, because

24 times m-reduction [i] would yield digital (9, 65, 71)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7⁶⁵, 71, F₇, 56) (dual of [71, 6, 57]-code), but
  - residual code [i] would yield linear OA(7⁹, 14, F₇, 8) (dual of [14, 5, 9]-code), but
    - â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]

(89−80, 89, 72)-Net in Base 7 — Upper bound on s

There is no (9, 89, 73)-net in base 7, because

24 times m-reduction [i] would yield (9, 65, 73)-net in base 7, but
- extracting embedded orthogonal array [i] would yield OA(7⁶⁵, 73, S₇, 56), but
  - the linear programming bound shows that MÂ â‰¥ 22 811215 366107 453768 378823 709022 673015 989035 549897 706388 838989 / 2 012613 > 7⁶⁵ [i]