MinT - Best Known (98−12, 98, s)-Nets in Base 5

Best Known (98−12, 98, s)-Nets in Base 5

(98−12, 98, 1398110)-Net over F₅ — Constructive and digital

Digital (86, 98, 1398110)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 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 (79, 91, 1398100)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁹¹, 1398100, F₅, 12, 12) (dual of [(1398100, 12), 16777109, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(5⁹¹, 8388600, F₅, 12) (dual of [8388600, 8388509, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁹¹, large, F₅, 12) (dual of [large, large−91, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(98−12, 98, 6822208)-Net over F₅ — Digital

Digital (86, 98, 6822208)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁸, 6822208, F₅, 12) (dual of [6822208, 6822110, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁹⁸, large, F₅, 12) (dual of [large, large−98, 13]-code), using
  - 7 times code embedding in larger space [i] based on linear OA(5⁹¹, large, F₅, 12) (dual of [large, large−91, 13]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(98−12, 98, large)-Net in Base 5 — Upper bound on s

There is no (86, 98, large)-net in base 5, because

10 times m-reduction [i] would yield (86, 88, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]