MinT - Best Known (109−20, 109, s)-Nets in Base 5

Best Known (109−20, 109, s)-Nets in Base 5

(109−20, 109, 1578)-Net over F₅ — Constructive and digital

Digital (89, 109, 1578)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (76, 96, 1562)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁹⁶, 1562, F₅, 20, 20) (dual of [(1562, 20), 31144, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(5⁹⁶, 15620, F₅, 20) (dual of [15620, 15524, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁹⁶, 15625, F₅, 20) (dual of [15625, 15529, 21]-code), using
        1 times truncation [i] based on linear OA(5⁹⁷, 15626, F₅, 21) (dual of [15626, 15529, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 15626Â | 5¹²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(109−20, 109, 20288)-Net over F₅ — Digital

Digital (89, 109, 20288)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−20, 109, large)-Net in Base 5 — Upper bound on s

There is no (89, 109, large)-net in base 5, because

18 times m-reduction [i] would yield (89, 91, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]