MinT - Best Known (100−6, 100, s)-Nets in Base 7

Best Known (100−6, 100, s)-Nets in Base 7

(100−6, 100, large)-Net over F₇ — Constructive and digital

Digital (94, 100, large)-net over F₇, using

2 times m-reduction [i] based on digital (94, 102, large)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (22, 26, 5764804)-net over F₇, using
    - net defined by OOA [i] based on linear OOA(7²⁶, 5764804, F₇, 4, 4) (dual of [(5764804, 4), 23059190, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(7²⁶, 5764804, F₇, 3, 4) (dual of [(5764804, 3), 17294386, 5]-NRT-code), using
        trace code [i] based on linear OOA(49¹³, 2882402, F₄₉, 3, 4) (dual of [(2882402, 3), 8647193, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(49¹³, 5764804, F₄₉, 4) (dual of [5764804, 5764791, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(49¹³, 5764805, F₄₉, 4) (dual of [5764805, 5764792, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(49¹³, 5764801, F₄₉, 4) (dual of [5764801, 5764788, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(49⁹, 5764801, F₄₉, 3) (dual of [5764801, 5764792, 4]-code or 5764801-cap in PG(8,49)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  2. digital (68, 76, 4194308)-net over F₇, using
    - (u,Â u+v)-construction [i] based on
      1. digital (0, 4, 8)-net over F₇, using
        net from sequence [i] based on digital (0, 7)-sequence over F₇, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
        the rational function field F₇(x) [i]
        Niederreiter sequence [i]
      2. digital (64, 72, 4194300)-net over F₇, using
        net defined by OOA [i] based on linear OOA(7⁷², 4194300, F₇, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(7⁷², 8388601, F₇, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(7⁷², 8388602, F₇, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
        trace code [i] based on linear OOA(49³⁶, 4194301, F₄₉, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(49³⁶, 8388602, F₄₉, 8) (dual of [8388602, 8388566, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(49³⁶, large, F₄₉, 8) (dual of [large, large−36, 9]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 25679568Â | 49⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(100−6, 100, large)-Net in Base 7 — Upper bound on s

There is no (94, 100, large)-net in base 7, because

4 times m-reduction [i] would yield (94, 96, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (100−6, 100, s)-Nets in Base 7

(100−6, 100, large)-Net over F7 — Constructive and digital

(100−6, 100, large)-Net in Base 7 — Upper bound on s

(100−6, 100, large)-Net over F₇ — Constructive and digital