MinT - Best Known (32−4, 32, s)-Nets in Base 7

Best Known (32−4, 32, s)-Nets in Base 7

(32−4, 32, 8388602)-Net over F₇ — Constructive and digital

Digital (28, 32, 8388602)-net over F₇, using

net defined by OOA [i] based on linear OOA(7³², 8388602, F₇, 4, 4) (dual of [(8388602, 4), 33554376, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(7³², 8388602, F₇, 3, 4) (dual of [(8388602, 3), 25165774, 5]-NRT-code), using
  - trace code [i] based on linear OOA(49¹⁶, 4194301, F₄₉, 3, 4) (dual of [(4194301, 3), 12582887, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(49¹⁶, 8388602, F₄₉, 4) (dual of [8388602, 8388586, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(49¹⁶, large, F₄₉, 4) (dual of [large, large−16, 5]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 25679568Â | 49⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(32−4, 32, large)-Net over F₇ — Digital

Digital (28, 32, large)-net over F₇, using

7⁶ times duplication [i] based on digital (22, 26, large)-net over F₇, using
- net defined by OOA [i] based on linear OOA(7²⁶, large, F₇, 4, 4), using
  - appending kth column [i] based on linear OOA(7²⁶, large, F₇, 3, 4), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²⁶, large, F₇, 4) (dual of [large, large−26, 5]-code), using
      - trace code [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]

(32−4, 32, large)-Net in Base 7 — Upper bound on s

There is no (28, 32, large)-net in base 7, because

2 times m-reduction [i] would yield (28, 30, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]