MinT - Best Known (86−12, 86, s)-Nets in Base 16

Best Known (86−12, 86, s)-Nets in Base 16

(86−12, 86, 2797569)-Net over F₁₆ — Constructive and digital

Digital (74, 86, 2797569)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (12, 18, 1369)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16¹⁸, 1369, F₁₆, 6, 6) (dual of [(1369, 6), 8196, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(16¹⁸, 4107, F₁₆, 6) (dual of [4107, 4089, 7]-code), using
      - constructionÂ X applied to C_e(5) âŠ‚ C_e(2) [i] based on
        linear OA(16¹⁶, 4096, F₁₆, 6) (dual of [4096, 4080, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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(16², 11, F₁₆, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,16)), using
        discarding factors / shortening the dual code based on linear OA(16², 16, F₁₆, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
        Reedâ€“Solomon code RS(14,16) [i]
2. digital (56, 68, 2796200)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(86−12, 86, large)-Net over F₁₆ — Digital

Digital (74, 86, large)-net over F₁₆, using

4 times m-reduction [i] based on digital (74, 90, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁰, large, F₁₆, 16) (dual of [large, large−90, 17]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(86−12, 86, large)-Net in Base 16 — Upper bound on s

There is no (74, 86, large)-net in base 16, because

10 times m-reduction [i] would yield (74, 76, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (86−12, 86, s)-Nets in Base 16

(86−12, 86, 2797569)-Net over F16 — Constructive and digital

(86−12, 86, large)-Net over F16 — Digital

(86−12, 86, large)-Net in Base 16 — Upper bound on s

(86−12, 86, 2797569)-Net over F₁₆ — Constructive and digital

(86−12, 86, large)-Net over F₁₆ — Digital