MinT - Best Known (5, 5+29, s)-Nets in Base 7

Best Known (5, 5+29, s)-Nets in Base 7

(5, 5+29, 13)-Net over F₇ — Constructive and digital

Digital (5, 34, 13)-net over F₇, using

net from sequence [i] based on digital (5, 12)-sequence over F₇, using
- Niederreiter sequence [i]

(5, 5+29, 24)-Net over F₇ — Digital

Digital (5, 34, 24)-net over F₇, using

t-expansion [i] based on digital (4, 34, 24)-net over F₇, using
- net from sequence [i] based on digital (4, 23)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 4 and N(F) ≥ 24, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(5, 5+29, 57)-Net over F₇ — Upper bound on s (digital)

There is no digital (5, 34, 58)-net over F₇, because

1 times m-reduction [i] would yield digital (5, 33, 58)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7³³, 58, F₇, 28) (dual of [58, 25, 29]-code), but
  - residual code [i] would yield OA(7⁵, 29, S₇, 4), but
    - the linear programming bound shows that MÂ â‰¥ 3 384381 / 197 > 7⁵ [i]

(5, 5+29, 75)-Net in Base 7 — Upper bound on s

There is no (5, 34, 76)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7³⁴, 76, S₇, 29), but
- the linear programming bound shows that MÂ â‰¥ 1104 767111 432500 930376 966448 543772 525848 489323 448063 923277 690120 / 18930 999823 588323 231167 214657 916817 > 7³⁴ [i]