MinT - Best Known (89, 100, s)-Nets in Base 9

Best Known (89, 100, s)-Nets in Base 9

(89, 100, 3362740)-Net over F₉ — Constructive and digital

Digital (89, 100, 3362740)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. 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
        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]
        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]
2. digital (71, 82, 3355440)-net over F₉, using
  - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁴¹, 1677720, F₈₁, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(81⁴¹, 8388601, F₈₁, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴¹, large, F₈₁, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(89, 100, large)-Net over F₉ — Digital

Digital (89, 100, large)-net over F₉, using

4 times m-reduction [i] based on digital (89, 104, large)-net over F₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(89, 100, large)-Net in Base 9 — Upper bound on s

There is no (89, 100, large)-net in base 9, because

9 times m-reduction [i] would yield (89, 91, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (89, 100, s)-Nets in Base 9

(89, 100, 3362740)-Net over F9 — Constructive and digital

(89, 100, large)-Net over F9 — Digital

(89, 100, large)-Net in Base 9 — Upper bound on s

(89, 100, 3362740)-Net over F₉ — Constructive and digital

(89, 100, large)-Net over F₉ — Digital