MinT - Best Known (3, 3+5, s)-Nets in Base 8

Best Known (3, 3+5, s)-Nets in Base 8

(3, 3+5, 28)-Net over F₈ — Constructive and digital

Digital (3, 8, 28)-net over F₈, using

8¹ times duplication [i] based on digital (2, 7, 28)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁷, 28, F₈, 5, 5) (dual of [(28, 5), 133, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
    - a code by Danev and Olsson [i]

(3, 3+5, 32)-Net over F₈ — Digital

Digital (3, 8, 32)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸, 32, F₈, 5) (dual of [32, 24, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸, 58, F₈, 5) (dual of [58, 50, 6]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
    - a code by Danev and Olsson [i]

(3, 3+5, 291)-Net in Base 8 — Upper bound on s

There is no (3, 8, 292)-net in base 8, because

1 times m-reduction [i] would yield (3, 7, 292)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 100211 > 8⁷ [i]
- extracting embedded orthogonal array [i] would yield OA(8⁷, 292, S₈, 4), but
  - the linear programming bound shows that MÂ â‰¥ 240314 880000 / 114211 > 8⁷ [i]