MinT - Best Known (4, 34, s)-Nets in Base 7

Best Known (4, 34, s)-Nets in Base 7

(4, 34, 12)-Net over F₇ — Constructive and digital

Digital (4, 34, 12)-net over F₇, using

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

(4, 34, 24)-Net over F₇ — Digital

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]

(4, 34, 36)-Net over F₇ — Upper bound on s (digital)

There is no digital (4, 34, 37)-net over F₇, because

2 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]

(4, 34, 40)-Net in Base 7 — Upper bound on s

There is no (4, 34, 41)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7³⁴, 41, S₇, 30), but
- the linear programming bound shows that MÂ â‰¥ 1620 965184 204779 600820 348721 410097 / 22971 > 7³⁴ [i]