MinT - Best Known (46−8, 46, s)-Nets in Base 16

Best Known (46−8, 46, s)-Nets in Base 16

(46−8, 46, 4194301)-Net over F₁₆ — Constructive and digital

Digital (38, 46, 4194301)-net over F₁₆, using

net defined by OOA [i] based on linear OOA(16⁴⁶, 4194301, F₁₆, 10, 8) (dual of [(4194301, 10), 41942964, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(16⁴⁶, large, F₁₆, 2, 8), using
  - 1 times NRT-code embedding in larger space [i] 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]

(46−8, 46, large)-Net over F₁₆ — Digital

Digital (38, 46, large)-net over F₁₆, using

16³ times duplication [i] based on digital (35, 43, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴³, large, F₁₆, 8) (dual of [large, large−43, 9]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(46−8, 46, large)-Net in Base 16 — Upper bound on s

There is no (38, 46, large)-net in base 16, because

6 times m-reduction [i] would yield (38, 40, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]