MinT - Best Known (85−12, 85, s)-Nets in Base 16

Best Known (85−12, 85, s)-Nets in Base 16

(85−12, 85, 2797568)-Net over F₁₆ — Constructive and digital

Digital (73, 85, 2797568)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (11, 17, 1368)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16¹⁷, 1368, F₁₆, 6, 6) (dual of [(1368, 6), 8191, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(16¹⁷, 4104, F₁₆, 6) (dual of [4104, 4087, 7]-code), using
      - constructionÂ X4 applied to C_e(5) âŠ‚ C_e(3) [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(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, 8, F₁₆, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,16)), using
        dual of repetition code with length 8 [i]
        linear OA(16¹, 8, F₁₆, 1) (dual of [8, 7, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(16¹, 16, F₁₆, 1) (dual of [16, 15, 2]-code), using
        Reedâ€“Solomon code RS(15,16) [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]

(85−12, 85, large)-Net over F₁₆ — Digital

Digital (73, 85, large)-net over F₁₆, using

t-expansion [i] based on digital (70, 85, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁵, large, F₁₆, 15) (dual of [large, large−85, 16]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(85−12, 85, large)-Net in Base 16 — Upper bound on s

There is no (73, 85, large)-net in base 16, because

10 times m-reduction [i] would yield (73, 75, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (85−12, 85, s)-Nets in Base 16

(85−12, 85, 2797568)-Net over F16 — Constructive and digital

(85−12, 85, large)-Net over F16 — Digital

(85−12, 85, large)-Net in Base 16 — Upper bound on s

(85−12, 85, 2797568)-Net over F₁₆ — Constructive and digital

(85−12, 85, large)-Net over F₁₆ — Digital