MinT - Best Known (238−6, 238, s)-Nets in Base 2

Best Known (238−6, 238, s)-Nets in Base 2

(238−6, 238, large)-Net over F₂ — Constructive and digital

Digital (232, 238, large)-net over F₂, using

2⁷ times duplication [i] based on digital (225, 231, large)-net over F₂, using
- t-expansion [i] based on digital (221, 231, large)-net over F₂, using
  - trace code for nets [i] based on digital (67, 77, 2796201)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁷⁷, 2796201, F₈, 15, 10) (dual of [(2796201, 15), 41942938, 11]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁷⁷, 5592403, F₈, 3, 10) (dual of [(5592403, 3), 16777132, 11]-NRT-code), using
        1 times NRT-code embedding in larger space [i] based on linear OOA(8⁷⁴, 5592402, F₈, 3, 10) (dual of [(5592402, 3), 16777132, 11]-NRT-code), using
        trace code [i] based on linear OOA(64³⁷, 2796201, F₆₄, 3, 10) (dual of [(2796201, 3), 8388566, 11]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(238−6, 238, large)-Net in Base 2 — Upper bound on s

There is no (232, 238, large)-net in base 2, because

4 times m-reduction [i] would yield (232, 234, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]