MinT - Best Known (41−35, 41, s)-Nets in Base 7

Best Known (41−35, 41, s)-Nets in Base 7

(41−35, 41, 14)-Net over F₇ — Constructive and digital

Digital (6, 41, 14)-net over F₇, using

net from sequence [i] based on digital (6, 13)-sequence over F₇, using
- Niederreiter sequence [i]

(41−35, 41, 24)-Net over F₇ — Digital

Digital (6, 41, 24)-net over F₇, using

t-expansion [i] based on digital (4, 41, 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]

(41−35, 41, 65)-Net over F₇ — Upper bound on s (digital)

There is no digital (6, 41, 66)-net over F₇, because

extracting embedded orthogonal array [i] would yield linear OA(7⁴¹, 66, F₇, 35) (dual of [66, 25, 36]-code), but
- residual code [i] would yield OA(7⁶, 30, S₇, 5), but
  - 1 times truncation [i] would yield OA(7⁵, 29, S₇, 4), but
    - the linear programming bound shows that MÂ â‰¥ 3 384381 / 197 > 7⁵ [i]

(41−35, 41, 76)-Net in Base 7 — Upper bound on s

There is no (6, 41, 77)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7⁴¹, 77, S₇, 35), but
- the linear programming bound shows that MÂ â‰¥ 1 261556 207333 721370 976650 036899 535102 800487 164084 585216 010033 036212 110407 / 25 232803 147664 626567 913594 841375 581005 > 7⁴¹ [i]