MinT - Best Known (82−9, 82, s)-Nets in Base 7

Best Known (82−9, 82, s)-Nets in Base 7

(82−9, 82, 4194300)-Net over F₇ — Constructive and digital

Digital (73, 82, 4194300)-net over F₇, using

net defined by OOA [i] based on linear OOA(7⁸², 4194300, F₇, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(7⁸², 8388601, F₇, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(7⁸², 8388602, F₇, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
    - trace code [i] based on linear OOA(49⁴¹, 4194301, F₄₉, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(49⁴¹, 8388602, F₄₉, 9) (dual of [8388602, 8388561, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁴¹, large, F₄₉, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 28247525Â | 49¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(82−9, 82, large)-Net over F₇ — Digital

Digital (73, 82, large)-net over F₇, using

t-expansion [i] based on digital (71, 82, large)-net over F₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸², large, F₇, 11) (dual of [large, large−82, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 20176803Â | 7⁹âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(82−9, 82, large)-Net in Base 7 — Upper bound on s

There is no (73, 82, large)-net in base 7, because

7 times m-reduction [i] would yield (73, 75, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]