MinT - Best Known (82−8, 82, s)-Nets in Base 7

Best Known (82−8, 82, s)-Nets in Base 7

(82−8, 82, 4194474)-Net over F₇ — Constructive and digital

Digital (74, 82, 4194474)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 10, 174)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7¹⁰, 174, F₇, 4, 4) (dual of [(174, 4), 686, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(7¹⁰, 174, F₇, 3, 4) (dual of [(174, 3), 512, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(7¹⁰, 348, F₇, 4) (dual of [348, 338, 5]-code), using
        constructionÂ XX applied to C₁Â =Â C([341,1]), C₂Â =Â C([0,2]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,1]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([341,2]) [i] based on
        linear OA(7⁷, 342, F₇, 3) (dual of [342, 335, 4]-code or 342-cap in PG(6,7)), using the primitive BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â {−1,0,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(7⁷, 342, F₇, 3) (dual of [342, 335, 4]-code or 342-cap in PG(6,7)), using the primitive expurgated narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(7¹⁰, 342, F₇, 4) (dual of [342, 332, 5]-code), using the primitive BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â {−1,0,1,2}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(7⁴, 342, F₇, 2) (dual of [342, 338, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(7⁰, 3, F₇, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
        linear OA(7⁰, 3, F₇, 0) (dual of [3, 3, 1]-code) (see above)
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]

(82−8, 82, large)-Net over F₇ — Digital

Digital (74, 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]

(82−8, 82, large)-Net in Base 7 — Upper bound on s

There is no (74, 82, large)-net in base 7, because

6 times m-reduction [i] would yield (74, 76, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (82−8, 82, s)-Nets in Base 7

(82−8, 82, 4194474)-Net over F7 — Constructive and digital

(82−8, 82, large)-Net over F7 — Digital

(82−8, 82, large)-Net in Base 7 — Upper bound on s

(82−8, 82, 4194474)-Net over F₇ — Constructive and digital

(82−8, 82, large)-Net over F₇ — Digital