MinT - Best Known (17−5, 17, s)-Nets in Base 16

Best Known (17−5, 17, s)-Nets in Base 16

(17−5, 17, 65553)-Net over F₁₆ — Constructive and digital

Digital (12, 17, 65553)-net over F₁₆, using

net defined by OOA [i] based on linear OOA(16¹⁷, 65553, F₁₆, 6, 5) (dual of [(65553, 6), 393301, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(16¹⁷, 65554, F₁₆, 2, 5) (dual of [(65554, 2), 131091, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(16³, 273, F₁₆, 2, 2) (dual of [(273, 2), 543, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(16³, 273, F₁₆, 2) (dual of [273, 270, 3]-code), using
        Hamming code H(3,16) [i]
    2. 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]

(17−5, 17, 130835)-Net over F₁₆ — Digital

Digital (12, 17, 130835)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁷, 130835, F₁₆, 5) (dual of [130835, 130818, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(16³, 273, F₁₆, 2) (dual of [273, 270, 3]-code), using
    - Hamming code H(3,16) [i]
  2. 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]

(17−5, 17, large)-Net in Base 16 — Upper bound on s

There is no (12, 17, large)-net in base 16, because

3 times m-reduction [i] would yield (12, 14, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]