MinT - Best Known (50−8, 50, s)-Nets in Base 16

Best Known (50−8, 50, s)-Nets in Base 16

(50−8, 50, 4194420)-Net over F₁₆ — Constructive and digital

Digital (42, 50, 4194420)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 120)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁶, 120, F₁₆, 4, 4) (dual of [(120, 4), 474, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(16⁶, 240, F₁₆, 4) (dual of [240, 234, 5]-code), using
      - 1 times truncation [i] based on linear OA(16⁷, 241, F₁₆, 5) (dual of [241, 234, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (36, 44, 4194300)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁴⁴, 4194300, F₁₆, 10, 8) (dual of [(4194300, 10), 41942956, 9]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(16⁴⁴, 8388601, F₁₆, 2, 8) (dual of [(8388601, 2), 16777158, 9]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁴⁴, 8388602, F₁₆, 2, 8) (dual of [(8388602, 2), 16777160, 9]-NRT-code), using
        trace code [i] based on linear OOA(256²², 4194301, F₂₅₆, 2, 8) (dual of [(4194301, 2), 8388580, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256²², 8388602, F₂₅₆, 8) (dual of [8388602, 8388580, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(256²², large, F₂₅₆, 8) (dual of [large, large−22, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(50−8, 50, large)-Net over F₁₆ — Digital

Digital (42, 50, large)-net over F₁₆, using

16¹ times duplication [i] based on digital (41, 49, large)-net over F₁₆, using
- t-expansion [i] based on digital (40, 49, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴⁹, large, F₁₆, 9) (dual of [large, large−49, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(50−8, 50, large)-Net in Base 16 — Upper bound on s

There is no (42, 50, large)-net in base 16, because

6 times m-reduction [i] would yield (42, 44, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (50−8, 50, s)-Nets in Base 16

(50−8, 50, 4194420)-Net over F16 — Constructive and digital

(50−8, 50, large)-Net over F16 — Digital

(50−8, 50, large)-Net in Base 16 — Upper bound on s

(50−8, 50, 4194420)-Net over F₁₆ — Constructive and digital

(50−8, 50, large)-Net over F₁₆ — Digital