MinT - Best Known (105−9, 105, s)-Nets in Base 3

Best Known (105−9, 105, s)-Nets in Base 3

(105−9, 105, 2099336)-Net over F₃ — Constructive and digital

Digital (96, 105, 2099336)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 15, 2186)-net over F₃, using
  - nets constructed from net-embeddable BCH codes [i]
2. digital (81, 90, 2097150)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁹⁰, 2097150, F₃, 9, 9) (dual of [(2097150, 9), 18874260, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(3⁹⁰, 8388601, F₃, 9) (dual of [8388601, 8388511, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁹⁰, large, F₃, 9) (dual of [large, large−90, 10]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(105−9, 105, large)-Net over F₃ — Digital

Digital (96, 105, large)-net over F₃, using

3⁵ times duplication [i] based on digital (91, 100, large)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁰, large, F₃, 9) (dual of [large, large−100, 10]-code), using
  - 10 times code embedding in larger space [i] based on linear OA(3⁹⁰, large, F₃, 9) (dual of [large, large−90, 10]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(105−9, 105, large)-Net in Base 3 — Upper bound on s

There is no (96, 105, large)-net in base 3, because

7 times m-reduction [i] would yield (96, 98, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]