MinT - Best Known (85, 85+10, s)-Nets in Base 4

Best Known (85, 85+10, s)-Nets in Base 4

(85, 85+10, 1677784)-Net over F₄ — Constructive and digital

Digital (85, 95, 1677784)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 10, 64)-net over F₄, using
  - linear code embedding found in a computer search [i]
2. digital (75, 85, 1677720)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁸⁵, 1677720, F₄, 10, 10) (dual of [(1677720, 10), 16777115, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(4⁸⁵, 8388600, F₄, 10) (dual of [8388600, 8388515, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁸⁵, large, F₄, 10) (dual of [large, large−85, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(85, 85+10, large)-Net over F₄ — Digital

Digital (85, 95, large)-net over F₄, using

4³ times duplication [i] based on digital (82, 92, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹², large, F₄, 10) (dual of [large, large−92, 11]-code), using
  - 7 times code embedding in larger space [i] based on linear OA(4⁸⁵, large, F₄, 10) (dual of [large, large−85, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(85, 85+10, large)-Net in Base 4 — Upper bound on s

There is no (85, 95, large)-net in base 4, because

8 times m-reduction [i] would yield (85, 87, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]