MinT - Best Known (26−7, 26, s)-Nets in Base 5

Best Known (26−7, 26, s)-Nets in Base 5

(26−7, 26, 1043)-Net over F₅ — Constructive and digital

Digital (19, 26, 1043)-net over F₅, using

net defined by OOA [i] based on linear OOA(5²⁶, 1043, F₅, 7, 7) (dual of [(1043, 7), 7275, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(5²⁶, 3130, F₅, 7) (dual of [3130, 3104, 8]-code), using
  - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
    1. linear OA(5²⁶, 3125, F₅, 7) (dual of [3125, 3099, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    2. linear OA(5²¹, 3125, F₅, 6) (dual of [3125, 3104, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
    3. linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 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]

(26−7, 26, 2033)-Net over F₅ — Digital

Digital (19, 26, 2033)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²⁶, 2033, F₅, 7) (dual of [2033, 2007, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(5²⁶, 3124, F₅, 7) (dual of [3124, 3098, 8]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(26−7, 26, 303438)-Net in Base 5 — Upper bound on s

There is no (19, 26, 303439)-net in base 5, because

1 times m-reduction [i] would yield (19, 25, 303439)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 298023 446587 357749 > 5²⁵ [i]