MinT - Best Known (7, 57, s)-Nets in Base 7

Best Known (7, 57, s)-Nets in Base 7

(7, 57, 15)-Net over F₇ — Constructive and digital

Digital (7, 57, 15)-net over F₇, using

net from sequence [i] based on digital (7, 14)-sequence over F₇, using
- Niederreiter sequence [i]

(7, 57, 32)-Net over F₇ — Digital

Digital (7, 57, 32)-net over F₇, using

net from sequence [i] based on digital (7, 31)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 7 and N(F) ≥ 32, using
  - an algebraic function field over â„¤₇ by Yeo [i]

(7, 57, 58)-Net over F₇ — Upper bound on s (digital)

There is no digital (7, 57, 59)-net over F₇, because

1 times m-reduction [i] would yield digital (7, 56, 59)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7⁵⁶, 59, F₇, 49) (dual of [59, 3, 50]-code), but
  - Griesmer bound [i]

(7, 57, 59)-Net in Base 7 — Upper bound on s

There is no (7, 57, 60)-net in base 7, because

2 times m-reduction [i] would yield (7, 55, 60)-net in base 7, but
- extracting embedded orthogonal array [i] would yield OA(7⁵⁵, 60, S₇, 48), but
  - the linear programming bound shows that MÂ â‰¥ 392948 425633 075727 327211 671891 403255 967254 645259 / 11 > 7⁵⁵ [i]