MinT - Best Known (94−12, 94, s)-Nets in Base 16

Best Known (94−12, 94, s)-Nets in Base 16

(94−12, 94, 3145727)-Net over F₁₆ — Constructive and digital

Digital (82, 94, 3145727)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (20, 26, 349527)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16²⁶, 349527, F₁₆, 6, 6) (dual of [(349527, 6), 2097136, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(16²⁶, 1048581, F₁₆, 6) (dual of [1048581, 1048555, 7]-code), using
      - constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(16²⁶, 1048576, F₁₆, 6) (dual of [1048576, 1048550, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(16²¹, 1048576, F₁₆, 5) (dual of [1048576, 1048555, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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 (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]

(94−12, 94, large)-Net over F₁₆ — Digital

Digital (82, 94, large)-net over F₁₆, using

t-expansion [i] based on digital (80, 94, large)-net over F₁₆, using
- 4 times m-reduction [i] based on digital (80, 98, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁸, large, F₁₆, 18) (dual of [large, large−98, 19]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(16⁹⁷, large, F₁₆, 18) (dual of [large, large−97, 19]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(94−12, 94, large)-Net in Base 16 — Upper bound on s

There is no (82, 94, large)-net in base 16, because

10 times m-reduction [i] would yield (82, 84, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (94−12, 94, s)-Nets in Base 16

(94−12, 94, 3145727)-Net over F16 — Constructive and digital

(94−12, 94, large)-Net over F16 — Digital

(94−12, 94, large)-Net in Base 16 — Upper bound on s

(94−12, 94, 3145727)-Net over F₁₆ — Constructive and digital

(94−12, 94, large)-Net over F₁₆ — Digital