MinT - Best Known (89−4, 89, s)-Nets in Base 7

Best Known (89−4, 89, s)-Nets in Base 7

(89−4, 89, large)-Net over F₇ — Constructive and digital

Digital (85, 89, large)-net over F₇, using

t-expansion [i] based on digital (84, 89, large)-net over F₇, using
- 3 times m-reduction [i] based on digital (84, 92, large)-net over F₇, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (7, 9, 2882402)-net over F₇, using
      - s-reduction based on digital (7, 9, 6725601)-net over F₇, using
        kÂ =Â 2 construction [i]
    2. digital (21, 25, 2882402)-net over F₇, using
      - s-reduction based on digital (21, 25, 2882404)-net over F₇, using
        net defined by OOA [i] based on linear OOA(7²⁵, 2882404, F₇, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(7²⁵, 2882404, F₇, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(7²⁵, 5764808, F₇, 4) (dual of [5764808, 5764783, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(7²⁵, 5764809, F₇, 4) (dual of [5764809, 5764784, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(7²⁵, 5764801, F₇, 4) (dual of [5764801, 5764776, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 7⁸âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(7¹⁷, 5764801, F₇, 3) (dual of [5764801, 5764784, 4]-code or 5764801-cap in PG(16,7)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 7⁸âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(7⁰, 8, F₇, 0) (dual of [8, 8, 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]
    3. digital (50, 58, 2882402)-net over F₇, using
      - trace code for nets [i] based on digital (21, 29, 1441201)-net over F₄₉, using
        net defined by OOA [i] based on linear OOA(49²⁹, 1441201, F₄₉, 8, 8) (dual of [(1441201, 8), 11529579, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(49²⁹, 5764804, F₄₉, 8) (dual of [5764804, 5764775, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(49²⁹, 5764805, F₄₉, 8) (dual of [5764805, 5764776, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(49²⁹, 5764801, F₄₉, 8) (dual of [5764801, 5764772, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(49²⁵, 5764801, F₄₉, 7) (dual of [5764801, 5764776, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [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]

(89−4, 89, large)-Net in Base 7 — Upper bound on s

There is no (85, 89, large)-net in base 7, because

2 times m-reduction [i] would yield (85, 87, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (89−4, 89, s)-Nets in Base 7

(89−4, 89, large)-Net over F7 — Constructive and digital

(89−4, 89, large)-Net in Base 7 — Upper bound on s

(89−4, 89, large)-Net over F₇ — Constructive and digital