MinT - Best Known (12, 12+5, s)-Nets in Base 9

Best Known (12, 12+5, s)-Nets in Base 9

(12, 12+5, 6571)-Net over F₉ — Constructive and digital

Digital (12, 17, 6571)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹⁷, 6571, F₉, 6, 5) (dual of [(6571, 6), 39409, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(9¹⁷, 6572, F₉, 2, 5) (dual of [(6572, 2), 13127, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(9³, 91, F₉, 2, 2) (dual of [(91, 2), 179, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(9³, 91, F₉, 2) (dual of [91, 88, 3]-code), using
        Hamming code H(3,9) [i]
    2. linear OOA(9¹⁴, 6481, F₉, 2, 5) (dual of [(6481, 2), 12948, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
        trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(12, 12+5, 13053)-Net over F₉ — Digital

Digital (12, 17, 13053)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁷, 13053, F₉, 5) (dual of [13053, 13036, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(9³, 91, F₉, 2) (dual of [91, 88, 3]-code), using
    - Hamming code H(3,9) [i]
  2. linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
    - trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
      - a code by Danev and Olsson [i]

(12, 12+5, 7609656)-Net in Base 9 — Upper bound on s

There is no (12, 17, 7609657)-net in base 9, because

1 times m-reduction [i] would yield (12, 16, 7609657)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1853 020514 308305 > 9¹⁶ [i]