MinT - Best Known (6, 42, s)-Nets in Base 7

Best Known (6, 42, s)-Nets in Base 7

(6, 42, 14)-Net over F₇ — Constructive and digital

Digital (6, 42, 14)-net over F₇, using

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

(6, 42, 24)-Net over F₇ — Digital

Digital (6, 42, 24)-net over F₇, using

t-expansion [i] based on digital (4, 42, 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]

(6, 42, 65)-Net over F₇ — Upper bound on s (digital)

There is no digital (6, 42, 66)-net over F₇, because

1 times m-reduction [i] would yield digital (6, 41, 66)-net over F₇, but
- 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]

(6, 42, 71)-Net in Base 7 — Upper bound on s

There is no (6, 42, 72)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7⁴², 72, S₇, 36), but
- the linear programming bound shows that MÂ â‰¥ 1317 858742 541027 651897 283715 012422 855235 377901 886670 269689 / 4192 095566 060516 873025 > 7⁴² [i]