MinT - Best Known (88, 99, s)-Nets in Base 9

Best Known (88, 99, s)-Nets in Base 9

(88, 99, 3362011)-Net over F₉ — Constructive and digital

Digital (88, 99, 3362011)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. 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
        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]
        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]

(88, 99, large)-Net over F₉ — Digital

Digital (88, 99, large)-net over F₉, using

9² times duplication [i] based on digital (86, 97, large)-net over F₉, using
- t-expansion [i] based on digital (83, 97, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(88, 99, large)-Net in Base 9 — Upper bound on s

There is no (88, 99, large)-net in base 9, because

9 times m-reduction [i] would yield (88, 90, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (88, 99, s)-Nets in Base 9

(88, 99, 3362011)-Net over F9 — Constructive and digital

(88, 99, large)-Net over F9 — Digital

(88, 99, large)-Net in Base 9 — Upper bound on s

(88, 99, 3362011)-Net over F₉ — Constructive and digital

(88, 99, large)-Net over F₉ — Digital