MinT - Best Known (13−4, 13, s)-Nets in Base 8

Best Known (13−4, 13, s)-Nets in Base 8

(13−4, 13, 4032)-Net over F₈ — Constructive and digital

Digital (9, 13, 4032)-net over F₈, using

1 times m-reduction [i] based on digital (9, 14, 4032)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁴, 4032, F₈, 5, 5) (dual of [(4032, 5), 20146, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(8¹⁴, 8065, F₈, 5) (dual of [8065, 8051, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
      - trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]

(13−4, 13, 8066)-Net over F₈ — Digital

Digital (9, 13, 8066)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹³, 8066, F₈, 4, 4) (dual of [(8066, 4), 32251, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8¹³, 8066, F₈, 3, 4) (dual of [(8066, 3), 24185, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³, 8066, F₈, 4) (dual of [8066, 8053, 5]-code), using
    - constructionÂ X with VarÅ¡amov bound [i] based on
      1. linear OA(8¹², 8064, F₈, 4) (dual of [8064, 8052, 5]-code), using
        trace code [i] based on linear OA(64⁶, 4032, F₆₄, 4) (dual of [4032, 4026, 5]-code), using
        1 times truncation [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
      2. linear OA(8¹², 8065, F₈, 3) (dual of [8065, 8053, 4]-code or 8065-cap in PG(11,8)), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8¹²Â > V_b^sâˆ’1(kâˆ’1)Â = 1593 047233 [i]
      3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 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, 149795)-Net in Base 8 — Upper bound on s

There is no (9, 13, 149796)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 549757 386739 > 8¹³ [i]