MinT - Best Known (35, 50, s)-Nets in Base 5

Best Known (35, 50, s)-Nets in Base 5

(35, 50, 252)-Net over F₅ — Constructive and digital

Digital (35, 50, 252)-net over F₅, using

trace code for nets [i] based on digital (10, 25, 126)-net over F₂₅, using
- net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
    - the Hermitian function field over F₂₅ [i]

(35, 50, 602)-Net over F₅ — Digital

Digital (35, 50, 602)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵⁰, 602, F₅, 15) (dual of [602, 552, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁵⁰, 635, F₅, 15) (dual of [635, 585, 16]-code), using
  - constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
    1. linear OA(5⁴⁹, 626, F₅, 15) (dual of [626, 577, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    2. linear OA(5⁴¹, 626, F₅, 13) (dual of [626, 585, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    3. linear OA(5¹, 9, F₅, 1) (dual of [9, 8, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(35, 50, 66011)-Net in Base 5 — Upper bound on s

There is no (35, 50, 66012)-net in base 5, because

1 times m-reduction [i] would yield (35, 49, 66012)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 17765 129535 133840 377738 403401 178865 > 5⁴⁹ [i]