MinT - Best Known (15−5, 15, s)-Nets in Base 8

Best Known (15−5, 15, s)-Nets in Base 8

Digital (10, 15, 4033)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁵, 4033, F₈, 5, 5) (dual of [(4033, 5), 20150, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8¹⁵, 8067, F₈, 5) (dual of [8067, 8052, 6]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
    - trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
      - a code by Danev and Olsson [i]

Digital (10, 15, 4252)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵, 4252, F₈, 5) (dual of [4252, 4237, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹⁵, 8067, F₈, 5) (dual of [8067, 8052, 6]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
    - trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
      - a code by Danev and Olsson [i]

There is no (10, 15, 423688)-net in base 8, because

1 times m-reduction [i] would yield (10, 14, 423688)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 398048 584917 > 8¹⁴ [i]