MinT - Best Known (90−9, 90, s)-Nets in Base 16

Best Known (90−9, 90, s)-Nets in Base 16

(90−9, 90, large)-Net over F₁₆ — Constructive and digital

Digital (81, 90, large)-net over F₁₆, using

1 times m-reduction [i] based on digital (81, 91, large)-net over F₁₆, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 10, 3355440)-net over F₁₆, using
    - s-reduction based on digital (7, 10, large)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16¹⁰, large, F₁₆, 3, 3), using
        appending kth column [i] based on linear OOA(16¹⁰, large, F₁₆, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16¹⁰, large, F₁₆, 3) (dual of [large, large−10, 4]-code), using
        product of projective cap with 16²-ovoid cap in AG(3,Â 16) [i] based on linear OA(16⁷, 66048, F₁₆, 3) (dual of [66048, 66041, 4]-code or 66048-cap in PG(6,16)), using
        product of two caps with tangent hyperplane [i]
  2. digital (20, 25, 3355440)-net over F₁₆, using
    - s-reduction based on digital (20, 25, 4194301)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16²⁵, 4194301, F₁₆, 5, 5) (dual of [(4194301, 5), 20971480, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(16²⁵, large, F₁₆, 5) (dual of [large, large−25, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
  3. digital (46, 56, 3355440)-net over F₁₆, using
    - trace code for nets [i] based on digital (18, 28, 1677720)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256²⁸, 1677720, F₂₅₆, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(256²⁸, 8388600, F₂₅₆, 10) (dual of [8388600, 8388572, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁸, large, F₂₅₆, 10) (dual of [large, large−28, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(90−9, 90, large)-Net in Base 16 — Upper bound on s

There is no (81, 90, large)-net in base 16, because

7 times m-reduction [i] would yield (81, 83, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (90−9, 90, s)-Nets in Base 16

(90−9, 90, large)-Net over F16 — Constructive and digital

(90−9, 90, large)-Net in Base 16 — Upper bound on s

(90−9, 90, large)-Net over F₁₆ — Constructive and digital