MinT - Best Known (30, 51, s)-Nets in Base 7

Best Known (30, 51, s)-Nets in Base 7

(30, 51, 108)-Net over F₇ — Constructive and digital

Digital (30, 51, 108)-net over F₇, using

1 times m-reduction [i] based on digital (30, 52, 108)-net over F₇, using
- trace code for nets [i] based on digital (4, 26, 54)-net over F₄₉, using
  - net from sequence [i] based on digital (4, 53)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(30, 51, 209)-Net over F₇ — Digital

Digital (30, 51, 209)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁵¹, 209, F₇, 21) (dual of [209, 158, 22]-code), using
- 21 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 6 times 0, 1, 13 times 0) [i] based on linear OA(7⁴⁹, 186, F₇, 21) (dual of [186, 137, 22]-code), using
  - constructionÂ X with VarÅ¡amov bound [i] based on
    1. linear OA(7⁴⁸, 184, F₇, 21) (dual of [184, 136, 22]-code), using
      - trace code [i] based on linear OA(49²⁴, 92, F₄₉, 21) (dual of [92, 68, 22]-code), using
        extended algebraic-geometric code AG_e(F,70P) [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92, using
        a curve from the manYPoints database [i]
    2. linear OA(7⁴⁸, 185, F₇, 20) (dual of [185, 137, 21]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 7⁴⁸Â > V_b^sâˆ’1(kâˆ’1)Â = 20980 321116 072652 872255 199308 869480 905537 [i]
    3. linear OA(7⁰, 1, F₇, 0) (dual of [1, 1, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(30, 51, 12679)-Net in Base 7 — Upper bound on s

There is no (30, 51, 12680)-net in base 7, because

1 times m-reduction [i] would yield (30, 50, 12680)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 799091 496126 536919 313267 343721 436779 343089 > 7⁵⁰ [i]