MinT - Best Known (72−11, 72, s)-Nets in Base 5

Best Known (72−11, 72, s)-Nets in Base 5

(72−11, 72, 78146)-Net over F₅ — Constructive and digital

Digital (61, 72, 78146)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 21)-net over F₅, using
  - linear code embedding found in a computer search [i]
2. digital (54, 65, 78125)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁶⁵, 78125, F₅, 11, 11) (dual of [(78125, 11), 859310, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁶⁵, 390626, F₅, 11) (dual of [390626, 390561, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 390626Â | 5¹⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(72−11, 72, 338673)-Net over F₅ — Digital

Digital (61, 72, 338673)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷², 338673, F₅, 11) (dual of [338673, 338601, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁷², 390647, F₅, 11) (dual of [390647, 390575, 12]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(5⁷, 21, F₅, 5) (dual of [21, 14, 6]-code), using
      - a code by Danev and Olsson [i]
    2. linear OA(5⁶⁵, 390626, F₅, 11) (dual of [390626, 390561, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 390626Â | 5¹⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(72−11, 72, large)-Net in Base 5 — Upper bound on s

There is no (61, 72, large)-net in base 5, because

9 times m-reduction [i] would yield (61, 63, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]