MinT - Best Known (25, 25+21, s)-Nets in Base 49

Best Known (25, 25+21, s)-Nets in Base 49

(25, 25+21, 344)-Net over F₄₉ — Constructive and digital

Digital (25, 46, 344)-net over F₄₉, using

t-expansion [i] based on digital (21, 46, 344)-net over F₄₉, using
- net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
    - the Hermitian function field over F₄₉ [i]

(25, 25+21, 1655)-Net over F₄₉ — Digital

Digital (25, 46, 1655)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁴⁶, 1655, F₄₉, 21) (dual of [1655, 1609, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(49⁴⁶, 2419, F₄₉, 21) (dual of [2419, 2373, 22]-code), using
  - constructionÂ X applied to C([0,10]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(49⁴¹, 2402, F₄₉, 21) (dual of [2402, 2361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402Â | 49⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    2. linear OA(49²⁹, 2402, F₄₉, 15) (dual of [2402, 2373, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402Â | 49⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    3. linear OA(49⁵, 17, F₄₉, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,49)), using
      - discarding factors / shortening the dual code based on linear OA(49⁵, 49, F₄₉, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
        Reedâ€“Solomon code RS(44,49) [i]

(25, 25+21, 3807298)-Net in Base 49 — Upper bound on s

There is no (25, 46, 3807299)-net in base 49, because

1 times m-reduction [i] would yield (25, 45, 3807299)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 11450 502851 140941 949455 724544 733721 503165 642043 658214 373235 033692 746665 450401 > 49⁴⁵ [i]