MinT - Best Known (43−8, 43, s)-Nets in Base 5

Best Known (43−8, 43, s)-Nets in Base 5

(43−8, 43, 19533)-Net over F₅ — Constructive and digital

Digital (35, 43, 19533)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁴³, 19533, F₅, 8, 8) (dual of [(19533, 8), 156221, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(5⁴³, 78132, F₅, 8) (dual of [78132, 78089, 9]-code), using
  - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
    1. linear OA(5⁴³, 78125, F₅, 8) (dual of [78125, 78082, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    2. linear OA(5³⁶, 78125, F₅, 7) (dual of [78125, 78089, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    3. linear OA(5⁰, 7, F₅, 0) (dual of [7, 7, 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]

(43−8, 43, 58469)-Net over F₅ — Digital

Digital (35, 43, 58469)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴³, 58469, F₅, 8) (dual of [58469, 58426, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁴³, 78125, F₅, 8) (dual of [78125, 78082, 9]-code), using
  - an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(43−8, 43, large)-Net in Base 5 — Upper bound on s

There is no (35, 43, large)-net in base 5, because

6 times m-reduction [i] would yield (35, 37, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]