MinT - Best Known (111−8, 111, s)-Nets in Base 9

Best Known (111−8, 111, s)-Nets in Base 9

(111−8, 111, large)-Net over F₉ — Constructive and digital

Digital (103, 111, large)-net over F₉, using

9¹³ times duplication [i] based on digital (90, 98, large)-net over F₉, using
- t-expansion [i] based on digital (89, 98, large)-net over F₉, using
  - trace code for nets [i] based on digital (40, 49, 4194381)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁴⁹, 4194381, F₈₁, 12, 9) (dual of [(4194381, 12), 50332523, 10]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(81⁴⁹, 4194382, F₈₁, 4, 9) (dual of [(4194382, 4), 16777479, 10]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(81³, 82, F₈₁, 4, 3) (dual of [(82, 4), 325, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;325,81) [i]
        linear OOA(81¹³, 2097150, F₈₁, 4, 4) (dual of [(2097150, 4), 8388587, 5]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81¹³, 4194301, F₈₁, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(81¹³, 8388602, F₈₁, 4) (dual of [8388602, 8388589, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹³, large, F₈₁, 4) (dual of [large, large−13, 5]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OOA(81³³, 2097150, F₈₁, 4, 9) (dual of [(2097150, 4), 8388567, 10]-NRT-code), using
        OOA 4-folding [i] based on linear OA(81³³, 8388600, F₈₁, 9) (dual of [8388600, 8388567, 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]

(111−8, 111, large)-Net in Base 9 — Upper bound on s

There is no (103, 111, large)-net in base 9, because

6 times m-reduction [i] would yield (103, 105, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (111−8, 111, s)-Nets in Base 9

(111−8, 111, large)-Net over F9 — Constructive and digital

(111−8, 111, large)-Net in Base 9 — Upper bound on s

(111−8, 111, large)-Net over F₉ — Constructive and digital