MinT - Best Known (51, 51+21, s)-Nets in Base 7

Best Known (51, 51+21, s)-Nets in Base 7

Digital (51, 72, 239)-net over F₇, using

net defined by OOA [i] based on linear OOA(7⁷², 239, F₇, 21, 21) (dual of [(239, 21), 4947, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(7⁷², 2391, F₇, 21) (dual of [2391, 2319, 22]-code), using
  - discarding factors / shortening the dual code based on linear OA(7⁷², 2400, F₇, 21) (dual of [2400, 2328, 22]-code), using
    - 1 times truncation [i] based on linear OA(7⁷³, 2401, F₇, 22) (dual of [2401, 2328, 23]-code), using
      - an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

Digital (51, 72, 1890)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁷², 1890, F₇, 21) (dual of [1890, 1818, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁷², 2400, F₇, 21) (dual of [2400, 2328, 22]-code), using
  - 1 times truncation [i] based on linear OA(7⁷³, 2401, F₇, 22) (dual of [2401, 2328, 23]-code), using
    - an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

There is no (51, 72, 755123)-net in base 7, because

1 times m-reduction [i] would yield (51, 71, 755123)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 004527 711347 240329 443749 112493 371125 418883 924333 362053 691501 > 7⁷¹ [i]