MinT - Best Known (50−21, 50, s)-Nets in Base 49

Best Known (50−21, 50, s)-Nets in Base 49

(50−21, 50, 344)-Net over F₄₉ — Constructive and digital

Digital (29, 50, 344)-net over F₄₉, using

t-expansion [i] based on digital (21, 50, 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]

(50−21, 50, 2997)-Net over F₄₉ — Digital

Digital (29, 50, 2997)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁵⁰, 2997, F₄₉, 21) (dual of [2997, 2947, 22]-code), using
- 585 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 62 times 0, 1, 154 times 0, 1, 330 times 0) [i] based on linear OA(49⁴¹, 2403, F₄₉, 21) (dual of [2403, 2362, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
    1. linear OA(49⁴¹, 2401, F₄₉, 21) (dual of [2401, 2360, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(49³⁹, 2401, F₄₉, 20) (dual of [2401, 2362, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [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]

(50−21, 50, large)-Net in Base 49 — Upper bound on s

There is no (29, 50, large)-net in base 49, because

19 times m-reduction [i] would yield (29, 31, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]