MinT - Best Known (165−11, 165, s)-Nets in Base 4

Best Known (165−11, 165, s)-Nets in Base 4

(165−11, 165, large)-Net over F₄ — Constructive and digital

Digital (154, 165, large)-net over F₄, using

4⁵ times duplication [i] based on digital (149, 160, large)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 37, 4194301)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4³⁷, 4194301, F₄, 5, 5) (dual of [(4194301, 5), 20971468, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(4³⁷, large, F₄, 5) (dual of [large, large−37, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
  2. digital (112, 123, 5033160)-net over F₄, using
    - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(165−11, 165, large)-Net in Base 4 — Upper bound on s

There is no (154, 165, large)-net in base 4, because

9 times m-reduction [i] would yield (154, 156, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]