MinT - Best Known (13−4, 13, s)-Nets in Base 9

Best Known (13−4, 13, s)-Nets in Base 9

(13−4, 13, 6480)-Net over F₉ — Constructive and digital

Digital (9, 13, 6480)-net over F₉, using

1 times m-reduction [i] based on digital (9, 14, 6480)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹⁴, 6480, F₉, 5, 5) (dual of [(6480, 5), 32386, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(9¹⁴, 12961, F₉, 5) (dual of [12961, 12947, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
      - trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(13−4, 13, 12962)-Net over F₉ — Digital

Digital (9, 13, 12962)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹³, 12962, F₉, 4, 4) (dual of [(12962, 4), 51835, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(9¹³, 12962, F₉, 3, 4) (dual of [(12962, 3), 38873, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³, 12962, F₉, 4) (dual of [12962, 12949, 5]-code), using
    - constructionÂ X with VarÅ¡amov bound [i] based on
      1. linear OA(9¹², 12960, F₉, 4) (dual of [12960, 12948, 5]-code), using
        trace code [i] based on linear OA(81⁶, 6480, F₈₁, 4) (dual of [6480, 6474, 5]-code), using
        1 times truncation [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]
      2. linear OA(9¹², 12961, F₉, 3) (dual of [12961, 12949, 4]-code or 12961-cap in PG(11,9)), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9¹²Â > V_b^sâˆ’1(kâˆ’1)Â = 5374 460161 [i]
      3. linear OA(9⁰, 1, F₉, 0) (dual of [1, 1, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(13−4, 13, 281838)-Net in Base 9 — Upper bound on s

There is no (9, 13, 281839)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 541876 629745 > 9¹³ [i]