MinT - Best Known (18−5, 18, s)-Nets in Base 16

Best Known (18−5, 18, s)-Nets in Base 16

(18−5, 18, 69649)-Net over F₁₆ — Constructive and digital

Digital (13, 18, 69649)-net over F₁₆, using

net defined by OOA [i] based on linear OOA(16¹⁸, 69649, F₁₆, 6, 5) (dual of [(69649, 6), 417876, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(16¹⁸, 69650, F₁₆, 2, 5) (dual of [(69650, 2), 139282, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(16⁴, 4369, F₁₆, 2, 2) (dual of [(4369, 2), 8734, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(16⁴, 4369, F₁₆, 2) (dual of [4369, 4365, 3]-code), using
        Hamming code H(4,16) [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]

(18−5, 18, 134931)-Net over F₁₆ — Digital

Digital (13, 18, 134931)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁸, 134931, F₁₆, 5) (dual of [134931, 134913, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(16⁴, 4369, F₁₆, 2) (dual of [4369, 4365, 3]-code), using
    - Hamming code H(4,16) [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]

(18−5, 18, large)-Net in Base 16 — Upper bound on s

There is no (13, 18, large)-net in base 16, because

3 times m-reduction [i] would yield (13, 15, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]