MinT - Best Known (89−11, 89, s)-Nets in Base 9

Best Known (89−11, 89, s)-Nets in Base 9

(89−11, 89, 3355476)-Net over F₉ — Constructive and digital

Digital (78, 89, 3355476)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 36)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁷, 36, F₉, 5, 5) (dual of [(36, 5), 173, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(9⁷, 73, F₉, 5) (dual of [73, 66, 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]

(89−11, 89, large)-Net over F₉ — Digital

Digital (78, 89, large)-net over F₉, using

t-expansion [i] based on digital (76, 89, large)-net over F₉, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸⁹, large, F₉, 13) (dual of [large, large−89, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(89−11, 89, large)-Net in Base 9 — Upper bound on s

There is no (78, 89, large)-net in base 9, because

9 times m-reduction [i] would yield (78, 80, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]