MinT - Best Known (20−5, 20, s)-Nets in Base 16

Best Known (20−5, 20, s)-Nets in Base 16

(20−5, 20, 195842)-Net over F₁₆ — Constructive and digital

Digital (15, 20, 195842)-net over F₁₆, using

net defined by OOA [i] based on linear OOA(16²⁰, 195842, F₁₆, 6, 5) (dual of [(195842, 6), 1175032, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(16²⁰, 195843, F₁₆, 2, 5) (dual of [(195843, 2), 391666, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(16¹, 65281, F₁₆, 2, 1) (dual of [(65281, 2), 130561, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16¹, s, F₁₆, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    2. linear OOA(16⁵, 65281, F₁₆, 2, 2) (dual of [(65281, 2), 130557, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁵, 69905, F₁₆, 2, 2) (dual of [(69905, 2), 139805, 3]-NRT-code), using
        appending kth column [i] based on linear OA(16⁵, 69905, F₁₆, 2) (dual of [69905, 69900, 3]-code), using
        Hamming code H(5,16) [i]
    3. linear OOA(16¹⁴, 65281, F₁₆, 2, 5) (dual of [(65281, 2), 130548, 6]-NRT-code), using
      - OOA 2-folding [i] 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]

(20−5, 20, 270372)-Net over F₁₆ — Digital

Digital (15, 20, 270372)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16²⁰, 270372, F₁₆, 5) (dual of [270372, 270352, 6]-code), using
- (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
  1. linear OA(16¹, 69905, F₁₆, 1) (dual of [69905, 69904, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]
  2. linear OA(16⁵, 69905, F₁₆, 2) (dual of [69905, 69900, 3]-code), using
    - Hamming code H(5,16) [i]
  3. 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]

(20−5, 20, large)-Net in Base 16 — Upper bound on s

There is no (15, 20, large)-net in base 16, because

3 times m-reduction [i] would yield (15, 17, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]