MinT - Best Known (37−3, 37, s)-Nets in Base 7

Best Known (37−3, 37, s)-Nets in Base 7

(37−3, 37, large)-Net over F₇ — Constructive and digital

Digital (34, 37, large)-net over F₇, using

7³ times duplication [i] based on digital (31, 34, large)-net over F₇, using
- t-expansion [i] based on digital (30, 34, large)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7³⁴, large, F₇, 4, 4), using
    - appending kth column [i] based on linear OOA(7³⁴, large, F₇, 3, 4), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(7², 8, F₇, 3, 2) (dual of [(8, 3), 22, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;22,7) [i]
        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]

(37−3, 37, large)-Net in Base 7 — Upper bound on s

There is no (34, 37, large)-net in base 7, because

1 times m-reduction [i] would yield (34, 36, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]