MinT - Best Known (81−8, 81, s)-Nets in Base 4

Best Known (81−8, 81, s)-Nets in Base 4

(81−8, 81, 2097321)-Net over F₄ — Constructive and digital

Digital (73, 81, 2097321)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 9, 171)-net over F₄, using
  - nets constructed from net-embeddable BCH codes [i]
  - linear code embedding found in a computer search [i]
2. digital (64, 72, 2097150)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁷², 2097150, F₄, 8, 8) (dual of [(2097150, 8), 16777128, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(4⁷², 8388600, F₄, 8) (dual of [8388600, 8388528, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷², large, F₄, 8) (dual of [large, large−72, 9]-code), using
        the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(81−8, 81, large)-Net over F₄ — Digital

Digital (73, 81, large)-net over F₄, using

t-expansion [i] based on digital (72, 81, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸¹, large, F₄, 9) (dual of [large, large−81, 10]-code), using
  - 8 times code embedding in larger space [i] based on linear OA(4⁷³, large, F₄, 9) (dual of [large, large−73, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(81−8, 81, large)-Net in Base 4 — Upper bound on s

There is no (73, 81, large)-net in base 4, because

6 times m-reduction [i] would yield (73, 75, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]