MinT - Best Known (20, 29, s)-Nets in Base 7

Best Known (20, 29, s)-Nets in Base 7

(20, 29, 601)-Net over F₇ — Constructive and digital

Digital (20, 29, 601)-net over F₇, using

net defined by OOA [i] based on linear OOA(7²⁹, 601, F₇, 9, 9) (dual of [(601, 9), 5380, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(7²⁹, 2405, F₇, 9) (dual of [2405, 2376, 10]-code), using
  - constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
    1. linear OA(7²⁹, 2401, F₇, 9) (dual of [2401, 2372, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    2. linear OA(7²⁵, 2401, F₇, 8) (dual of [2401, 2376, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(7⁰, 4, F₇, 0) (dual of [4, 4, 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]

(20, 29, 1349)-Net over F₇ — Digital

Digital (20, 29, 1349)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²⁹, 1349, F₇, 9) (dual of [1349, 1320, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(7²⁹, 2401, F₇, 9) (dual of [2401, 2372, 10]-code), using
  - an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(20, 29, 303797)-Net in Base 7 — Upper bound on s

There is no (20, 29, 303798)-net in base 7, because

1 times m-reduction [i] would yield (20, 28, 303798)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 459987 246736 278522 844969 > 7²⁸ [i]