MinT - Best Known (76, 86, s)-Nets in Base 5

Best Known (76, 86, s)-Nets in Base 5

(76, 86, 1677730)-Net over F₅ — Constructive and digital

Digital (76, 86, 1677730)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [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]

(76, 86, large)-Net over F₅ — Digital

Digital (76, 86, large)-net over F₅, using

5⁵ times duplication [i] based on digital (71, 81, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸¹, large, F₅, 10) (dual of [large, large−81, 11]-code), using
  - strength reduction [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]

(76, 86, large)-Net in Base 5 — Upper bound on s

There is no (76, 86, large)-net in base 5, because

8 times m-reduction [i] would yield (76, 78, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]