MinT - Best Known (90−10, 90, s)-Nets in Base 5

Best Known (90−10, 90, s)-Nets in Base 5

(90−10, 90, 1677788)-Net over F₅ — Constructive and digital

Digital (80, 90, 1677788)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 10, 68)-net over F₅, using
  - linear code embedding found in a computer search [i]
2. digital (70, 80, 1677720)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸⁰, 1677720, F₅, 10, 10) (dual of [(1677720, 10), 16777120, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(5⁸⁰, 8388600, F₅, 10) (dual of [8388600, 8388520, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁸⁰, large, F₅, 10) (dual of [large, large−80, 11]-code), using
        the primitive narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(90−10, 90, large)-Net over F₅ — Digital

Digital (80, 90, large)-net over F₅, using

t-expansion [i] based on digital (79, 90, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁰, large, F₅, 11) (dual of [large, large−90, 12]-code), using
  - 9 times code embedding in larger space [i] based on linear OA(5⁸¹, large, F₅, 11) (dual of [large, large−81, 12]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(90−10, 90, large)-Net in Base 5 — Upper bound on s

There is no (80, 90, large)-net in base 5, because

8 times m-reduction [i] would yield (80, 82, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]