MinT - Best Known (174, 174+50, s)-Nets in Base 4

Best Known (174, 174+50, s)-Nets in Base 4

(174, 174+50, 1052)-Net over F₄ — Constructive and digital

Digital (174, 224, 1052)-net over F₄, using

trace code for nets [i] based on digital (6, 56, 263)-net over F₂₅₆, using
- net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(174, 174+50, 3877)-Net over F₄ — Digital

Digital (174, 224, 3877)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²⁴, 3877, F₄, 50) (dual of [3877, 3653, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(4²²⁴, 4103, F₄, 50) (dual of [4103, 3879, 51]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²²³, 4102, F₄, 50) (dual of [4102, 3879, 51]-code), using
    - constructionÂ X applied to C_e(49) âŠ‚ C_e(48) [i] based on
      1. linear OA(4²²³, 4096, F₄, 50) (dual of [4096, 3873, 51]-code), using an extension C_e(49) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,49], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 50 [i]
      2. linear OA(4²¹⁷, 4096, F₄, 49) (dual of [4096, 3879, 50]-code), using an extension C_e(48) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [i]
      3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(174, 174+50, 841301)-Net in Base 4 — Upper bound on s

There is no (174, 224, 841302)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 726 852230 797814 428456 182015 357375 821913 402888 942616 951111 898074 463805 796555 018721 028487 620005 173113 183480 095140 900034 298903 583903 122760 > 4²²⁴ [i]