MinT - Best Known (24, 24+25, s)-Nets in Base 49

Best Known (24, 24+25, s)-Nets in Base 49

(24, 24+25, 344)-Net over F₄₉ — Constructive and digital

Digital (24, 49, 344)-net over F₄₉, using

t-expansion [i] based on digital (21, 49, 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]

(24, 24+25, 801)-Net over F₄₉ — Digital

Digital (24, 49, 801)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(49⁴⁹, 801, F₄₉, 3, 25) (dual of [(801, 3), 2354, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(49⁴⁹, 2403, F₄₉, 25) (dual of [2403, 2354, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(23) [i] based on
    1. linear OA(49⁴⁹, 2401, F₄₉, 25) (dual of [2401, 2352, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(49⁴⁷, 2401, F₄₉, 24) (dual of [2401, 2354, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    3. linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(24, 24+25, 635185)-Net in Base 49 — Upper bound on s

There is no (24, 49, 635186)-net in base 49, because

1 times m-reduction [i] would yield (24, 48, 635186)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 1347 150353 274846 445565 431768 684635 742695 684444 015083 085771 765940 348173 974099 859073 > 49⁴⁸ [i]