MinT - Best Known (69−65, 69, s)-Nets in Base 7

Best Known (69−65, 69, s)-Nets in Base 7

(69−65, 69, 12)-Net over F₇ — Constructive and digital

Digital (4, 69, 12)-net over F₇, using

net from sequence [i] based on digital (4, 11)-sequence over F₇, using
- Niederreiter sequence [i]

(69−65, 69, 24)-Net over F₇ — Digital

Digital (4, 69, 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]

(69−65, 69, 36)-Net over F₇ — Upper bound on s (digital)

There is no digital (4, 69, 37)-net over F₇, because

37 times m-reduction [i] would yield digital (4, 32, 37)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7³², 37, F₇, 28) (dual of [37, 5, 29]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(7³¹, 33, S₇, 28), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 5522 138371 219603 231526 496005 / 29 > 7³¹ [i]
    2. OA(7⁵, 37, S₇, 4), but
      - discarding factors would yield OA(7⁵, 31, S₇, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 16927 > 7⁵ [i]

(69−65, 69, 39)-Net in Base 7 — Upper bound on s

There is no (4, 69, 40)-net in base 7, because

34 times m-reduction [i] would yield (4, 35, 40)-net in base 7, but
- extracting embedded orthogonal array [i] would yield OA(7³⁵, 40, S₇, 31), but
  - the linear programming bound shows that MÂ â‰¥ 18 562115 921017 574302 453163 671207 / 37 > 7³⁵ [i]