MinT - Best Known (13−4, 13, s)-Nets in Base 16

Best Known (13−4, 13, s)-Nets in Base 16

(13−4, 13, 65280)-Net over F₁₆ — Constructive and digital

Digital (9, 13, 65280)-net over F₁₆, using

1 times m-reduction [i] based on digital (9, 14, 65280)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16¹⁴, 65280, F₁₆, 5, 5) (dual of [(65280, 5), 326386, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(16¹⁴, 130561, F₁₆, 5) (dual of [130561, 130547, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(16¹⁴, 130562, F₁₆, 5) (dual of [130562, 130548, 6]-code), using
      - trace code [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]

(13−4, 13, 130562)-Net over F₁₆ — Digital

Digital (9, 13, 130562)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹³, 130562, F₁₆, 4) (dual of [130562, 130549, 5]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(16¹², 130560, F₁₆, 4) (dual of [130560, 130548, 5]-code), using
    - trace code [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
      - 1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
  2. linear OA(16¹², 130561, F₁₆, 3) (dual of [130561, 130549, 4]-code or 130561-cap in PG(11,16)), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 16¹²Â > V_b^sâˆ’1(kâˆ’1)Â = 1 917652 550401 [i]
  3. linear OA(16⁰, 1, F₁₆, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 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, 6327083)-Net in Base 16 — Upper bound on s

There is no (9, 13, 6327084)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 4503 599995 203271 > 16¹³ [i]