MinT - Best Known (58−15, 58, s)-Nets in Base 8

Best Known (58−15, 58, s)-Nets in Base 8

Digital (43, 58, 1170)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵⁸, 1170, F₈, 15, 15) (dual of [(1170, 15), 17492, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8⁵⁸, 8191, F₈, 15) (dual of [8191, 8133, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁵⁸, 8194, F₈, 15) (dual of [8194, 8136, 16]-code), using
    - trace code [i] based on linear OA(64²⁹, 4097, F₆₄, 15) (dual of [4097, 4068, 16]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

Digital (43, 58, 7372)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁸, 7372, F₈, 15) (dual of [7372, 7314, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵⁸, 8194, F₈, 15) (dual of [8194, 8136, 16]-code), using
  - trace code [i] based on linear OA(64²⁹, 4097, F₆₄, 15) (dual of [4097, 4068, 16]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

There is no (43, 58, large)-net in base 8, because

13 times m-reduction [i] would yield (43, 45, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]