MinT - Best Known (54−10, 54, s)-Nets in Base 5

Best Known (54−10, 54, s)-Nets in Base 5

(54−10, 54, 3135)-Net over F₅ — Constructive and digital

Digital (44, 54, 3135)-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 (38, 48, 3125)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁴⁸, 3125, F₅, 10, 10) (dual of [(3125, 10), 31202, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(5⁴⁸, 15625, F₅, 10) (dual of [15625, 15577, 11]-code), using
      - 1 times truncation [i] based on linear OA(5⁴⁹, 15626, F₅, 11) (dual of [15626, 15577, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 15626Â | 5¹²âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(54−10, 54, 16204)-Net over F₅ — Digital

Digital (44, 54, 16204)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−10, 54, large)-Net in Base 5 — Upper bound on s

There is no (44, 54, large)-net in base 5, because

8 times m-reduction [i] would yield (44, 46, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]