MinT - Best Known (54−8, 54, s)-Nets in Base 16

Best Known (54−8, 54, s)-Nets in Base 16

(54−8, 54, 4196349)-Net over F₁₆ — Constructive and digital

Digital (46, 54, 4196349)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 10, 2049)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16¹⁰, 2049, F₁₆, 4, 4) (dual of [(2049, 4), 8186, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(16¹⁰, 4098, F₁₆, 4) (dual of [4098, 4088, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹⁰, 4099, F₁₆, 4) (dual of [4099, 4089, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(16¹⁰, 4096, F₁₆, 4) (dual of [4096, 4086, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(16⁷, 4096, F₁₆, 3) (dual of [4096, 4089, 4]-code or 4096-cap in PG(6,16)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(16⁰, 3, F₁₆, 0) (dual of [3, 3, 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]

(54−8, 54, large)-Net over F₁₆ — Digital

Digital (46, 54, large)-net over F₁₆, using

t-expansion [i] based on digital (45, 54, large)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (45, 55, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁵, large, F₁₆, 10) (dual of [large, large−55, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(54−8, 54, large)-Net in Base 16 — Upper bound on s

There is no (46, 54, large)-net in base 16, because

6 times m-reduction [i] would yield (46, 48, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (54−8, 54, s)-Nets in Base 16

(54−8, 54, 4196349)-Net over F16 — Constructive and digital

(54−8, 54, large)-Net over F16 — Digital

(54−8, 54, large)-Net in Base 16 — Upper bound on s

(54−8, 54, 4196349)-Net over F₁₆ — Constructive and digital

(54−8, 54, large)-Net over F₁₆ — Digital