MinT - Best Known (20, 20+9, s)-Nets in Base 5

Best Known (20, 20+9, s)-Nets in Base 5

(20, 20+9, 157)-Net over F₅ — Constructive and digital

Digital (20, 29, 157)-net over F₅, using

net defined by OOA [i] based on linear OOA(5²⁹, 157, F₅, 9, 9) (dual of [(157, 9), 1384, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(5²⁹, 629, F₅, 9) (dual of [629, 600, 10]-code), using
  - constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
    1. linear OA(5²⁹, 625, F₅, 9) (dual of [625, 596, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    2. linear OA(5²⁵, 625, F₅, 8) (dual of [625, 600, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(5⁰, 4, F₅, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(20, 20+9, 524)-Net over F₅ — Digital

Digital (20, 29, 524)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²⁹, 524, F₅, 9) (dual of [524, 495, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(5²⁹, 624, F₅, 9) (dual of [624, 595, 10]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(20, 20+9, 43227)-Net in Base 5 — Upper bound on s

There is no (20, 29, 43228)-net in base 5, because

1 times m-reduction [i] would yield (20, 28, 43228)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 37 255445 029650 964225 > 5²⁸ [i]