MinT - Best Known (37, 37+13, s)-Nets in Base 8

Best Known (37, 37+13, s)-Nets in Base 8

Digital (37, 50, 1365)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵⁰, 1365, F₈, 13, 13) (dual of [(1365, 13), 17695, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8⁵⁰, 8191, F₈, 13) (dual of [8191, 8141, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁵⁰, 8194, F₈, 13) (dual of [8194, 8144, 14]-code), using
    - trace code [i] based on linear OA(64²⁵, 4097, F₆₄, 13) (dual of [4097, 4072, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

Digital (37, 50, 7386)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁰, 7386, F₈, 13) (dual of [7386, 7336, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵⁰, 8194, F₈, 13) (dual of [8194, 8144, 14]-code), using
  - trace code [i] based on linear OA(64²⁵, 4097, F₆₄, 13) (dual of [4097, 4072, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

There is no (37, 50, large)-net in base 8, because

11 times m-reduction [i] would yield (37, 39, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]