MinT - Best Known (77, 88, s)-Nets in Base 5

Best Known (77, 88, s)-Nets in Base 5

(77, 88, 1677741)-Net over F₅ — Constructive and digital

Digital (77, 88, 1677741)-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 (70, 81, 1677720)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸¹, 1677720, F₅, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁸¹, 8388601, F₅, 11) (dual of [8388601, 8388520, 12]-code), using
      - discarding factors / shortening the dual code 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]

(77, 88, 5921082)-Net over F₅ — Digital

Digital (77, 88, 5921082)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁸, 5921082, F₅, 11) (dual of [5921082, 5920994, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸⁸, large, F₅, 11) (dual of [large, large−88, 12]-code), using
  - 7 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]

(77, 88, large)-Net in Base 5 — Upper bound on s

There is no (77, 88, large)-net in base 5, because

9 times m-reduction [i] would yield (77, 79, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]