MinT - Best Known (50−16, 50, s)-Nets in Base 4

Best Known (50−16, 50, s)-Nets in Base 4

(50−16, 50, 195)-Net over F₄ — Constructive and digital

Digital (34, 50, 195)-net over F₄, using

1 times m-reduction [i] based on digital (34, 51, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 17, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(50−16, 50, 248)-Net over F₄ — Digital

Digital (34, 50, 248)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁰, 248, F₄, 16) (dual of [248, 198, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵⁰, 265, F₄, 16) (dual of [265, 215, 17]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [i] based on
    1. linear OA(4⁴⁹, 256, F₄, 17) (dual of [256, 207, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(4⁴¹, 256, F₄, 14) (dual of [256, 215, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(4¹, 9, F₄, 1) (dual of [9, 8, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(50−16, 50, 7262)-Net in Base 4 — Upper bound on s

There is no (34, 50, 7263)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 268602 419286 252192 540073 152645 > 4⁵⁰ [i]