MinT - Best Known (60−8, 60, s)-Nets in Base 16

Best Known (60−8, 60, s)-Nets in Base 16

(60−8, 60, 4718590)-Net over F₁₆ — Constructive and digital

Digital (52, 60, 4718590)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (12, 16, 524290)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16¹⁶, 524290, F₁₆, 4, 4) (dual of [(524290, 4), 2097144, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(16¹⁶, 1048580, F₁₆, 4) (dual of [1048580, 1048564, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹⁶, 1048581, F₁₆, 4) (dual of [1048581, 1048565, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(16¹⁶, 1048576, F₁₆, 4) (dual of [1048576, 1048560, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(16¹¹, 1048576, F₁₆, 3) (dual of [1048576, 1048565, 4]-code or 1048576-cap in PG(10,16)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(16⁰, 5, F₁₆, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 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]

(60−8, 60, large)-Net over F₁₆ — Digital

Digital (52, 60, large)-net over F₁₆, using

t-expansion [i] based on digital (50, 60, large)-net over F₁₆, using
- 1 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]

(60−8, 60, large)-Net in Base 16 — Upper bound on s

There is no (52, 60, large)-net in base 16, because

6 times m-reduction [i] would yield (52, 54, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (60−8, 60, s)-Nets in Base 16

(60−8, 60, 4718590)-Net over F16 — Constructive and digital

(60−8, 60, large)-Net over F16 — Digital

(60−8, 60, large)-Net in Base 16 — Upper bound on s

(60−8, 60, 4718590)-Net over F₁₆ — Constructive and digital

(60−8, 60, large)-Net over F₁₆ — Digital