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

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

(89−9, 89, 6585792)-Net over F₉ — Constructive and digital

Digital (80, 89, 6585792)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (19, 23, 2391492)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9²³, 2391492, F₉, 4, 4) (dual of [(2391492, 4), 9565945, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(9²³, 2391492, F₉, 3, 4) (dual of [(2391492, 3), 7174453, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(9²³, 4782984, F₉, 4) (dual of [4782984, 4782961, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(1) [i] based on
        linear OA(9²², 4782969, F₉, 4) (dual of [4782969, 4782947, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(9⁸, 4782969, F₉, 2) (dual of [4782969, 4782961, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
        linear OA(9¹, 15, F₉, 1) (dual of [15, 14, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (57, 66, 4194300)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁶⁶, 4194300, F₉, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(9⁶⁶, 8388601, F₉, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9⁶⁶, 8388602, F₉, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
        trace code [i] based on linear OOA(81³³, 4194301, F₈₁, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81³³, 8388602, F₈₁, 9) (dual of [8388602, 8388569, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(81³³, large, F₈₁, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(89−9, 89, large)-Net over F₉ — Digital

Digital (80, 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]

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

There is no (80, 89, large)-net in base 9, because

7 times m-reduction [i] would yield (80, 82, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

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

(89−9, 89, 6585792)-Net over F9 — Constructive and digital

(89−9, 89, large)-Net over F9 — Digital

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

(89−9, 89, 6585792)-Net over F₉ — Constructive and digital

(89−9, 89, large)-Net over F₉ — Digital