MinT - Best Known (88−10, 88, s)-Nets in Base 16

Best Known (88−10, 88, s)-Nets in Base 16

(88−10, 88, 6776928)-Net over F₁₆ — Constructive and digital

Digital (78, 88, 6776928)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 7, 66048)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁷, 66048, F₁₆, 3, 3) (dual of [(66048, 3), 198137, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(16⁷, 66048, F₁₆, 2, 3) (dual of [(66048, 2), 132089, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [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]

(88−10, 88, large)-Net over F₁₆ — Digital

Digital (78, 88, large)-net over F₁₆, using

t-expansion [i] based on digital (75, 88, large)-net over F₁₆, using
- 4 times m-reduction [i] based on digital (75, 92, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹², large, F₁₆, 17) (dual of [large, large−92, 18]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(16⁹¹, large, F₁₆, 17) (dual of [large, large−91, 18]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(88−10, 88, large)-Net in Base 16 — Upper bound on s

There is no (78, 88, large)-net in base 16, because

8 times m-reduction [i] would yield (78, 80, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (88−10, 88, s)-Nets in Base 16

(88−10, 88, 6776928)-Net over F16 — Constructive and digital

(88−10, 88, large)-Net over F16 — Digital

(88−10, 88, large)-Net in Base 16 — Upper bound on s

(88−10, 88, 6776928)-Net over F₁₆ — Constructive and digital

(88−10, 88, large)-Net over F₁₆ — Digital