MinT - Best Known (257−8, 257, s)-Nets in Base 2

Best Known (257−8, 257, s)-Nets in Base 2

(257−8, 257, large)-Net over F₂ — Constructive and digital

Digital (249, 257, large)-net over F₂, using

2² times duplication [i] based on digital (247, 255, large)-net over F₂, using
- t-expansion [i] based on digital (244, 255, large)-net over F₂, using
  - trace code for nets [i] based on digital (74, 85, 2796201)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁸⁵, 2796201, F₈, 15, 11) (dual of [(2796201, 15), 41942930, 12]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁸⁵, 5592403, F₈, 3, 11) (dual of [(5592403, 3), 16777124, 12]-NRT-code), using
        1 times NRT-code embedding in larger space [i] based on linear OOA(8⁸², 5592402, F₈, 3, 11) (dual of [(5592402, 3), 16777124, 12]-NRT-code), using
        trace code [i] based on linear OOA(64⁴¹, 2796201, F₆₄, 3, 11) (dual of [(2796201, 3), 8388562, 12]-NRT-code), using
        OOA 3-folding [i] 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]

(257−8, 257, large)-Net in Base 2 — Upper bound on s

There is no (249, 257, large)-net in base 2, because

6 times m-reduction [i] would yield (249, 251, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]