MinT - Best Known (50, 58, s)-Nets in Base 16

Best Known (50, 58, s)-Nets in Base 16

(50, 58, 4259838)-Net over F₁₆ — Constructive and digital

Digital (50, 58, 4259838)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (10, 14, 65538)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16¹⁴, 65538, F₁₆, 4, 4) (dual of [(65538, 4), 262138, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(16¹⁴, 131076, F₁₆, 4) (dual of [131076, 131062, 5]-code), using
      - trace code [i] based on linear OA(256⁷, 65538, F₂₅₆, 4) (dual of [65538, 65531, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(256⁷, 65536, F₂₅₆, 4) (dual of [65536, 65529, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(256⁵, 65536, F₂₅₆, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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, 58, large)-Net over F₁₆ — Digital

Digital (50, 58, large)-net over F₁₆, using

3 times m-reduction [i] based on digital (50, 61, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶¹, large, F₁₆, 11) (dual of [large, large−61, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(50, 58, large)-Net in Base 16 — Upper bound on s

There is no (50, 58, large)-net in base 16, because

6 times m-reduction [i] would yield (50, 52, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (50, 58, s)-Nets in Base 16

(50, 58, 4259838)-Net over F16 — Constructive and digital

(50, 58, large)-Net over F16 — Digital

(50, 58, large)-Net in Base 16 — Upper bound on s

(50, 58, 4259838)-Net over F₁₆ — Constructive and digital

(50, 58, large)-Net over F₁₆ — Digital