MinT - Best Known (18−5, 18, s)-Nets in Base 9

Best Known (18−5, 18, s)-Nets in Base 9

(18−5, 18, 7300)-Net over F₉ — Constructive and digital

Digital (13, 18, 7300)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹⁸, 7300, F₉, 6, 5) (dual of [(7300, 6), 43782, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(9¹⁸, 7301, F₉, 2, 5) (dual of [(7301, 2), 14584, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(9⁴, 820, F₉, 2, 2) (dual of [(820, 2), 1636, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(9⁴, 820, F₉, 2) (dual of [820, 816, 3]-code), using
        Hamming code H(4,9) [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]

(18−5, 18, 13782)-Net over F₉ — Digital

Digital (13, 18, 13782)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁸, 13782, F₉, 5) (dual of [13782, 13764, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(9⁴, 820, F₉, 2) (dual of [820, 816, 3]-code), using
    - Hamming code H(4,9) [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]

(18−5, 18, large)-Net in Base 9 — Upper bound on s

There is no (13, 18, large)-net in base 9, because

3 times m-reduction [i] would yield (13, 15, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]