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

Best Known (4, 4+∞, s)-Nets in Base 7

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

Digital (4, m, 12)-net over F₇ for arbitrarily large m, using

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

(4, 4+∞, 24)-Net over F₇ — Digital

Digital (4, m, 24)-net over F₇ for arbitrarily large m, 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, 4+∞, 36)-Net over F₇ — Upper bound on s (digital)

There is no digital (4, m, 37)-net over F₇ for arbitrarily large m, because

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, 4+∞, 38)-Net in Base 7 — Upper bound on s

There is no (4, m, 39)-net in base 7 for arbitrarily large m, because

m-reduction [i] would yield (4, 74, 39)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7⁷⁴, 39, S₇, 2, 70), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 30665 141124 411175 854056 235783 580684 649065 396053 855752 734074 479561 / 71 > 7⁷⁴ [i]