MinT - Best Known (168, 168+47, s)-Nets in Base 4

Best Known (168, 168+47, s)-Nets in Base 4

(168, 168+47, 1052)-Net over F₄ — Constructive and digital

Digital (168, 215, 1052)-net over F₄, using

1 times m-reduction [i] based on digital (168, 216, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 54, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(168, 168+47, 4138)-Net over F₄ — Digital

Digital (168, 215, 4138)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁵, 4138, F₄, 47) (dual of [4138, 3923, 48]-code), using
- 32 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 8 times 0, 1, 19 times 0) [i] based on linear OA(4²¹¹, 4102, F₄, 47) (dual of [4102, 3891, 48]-code), using
  - constructionÂ X applied to C_e(46) âŠ‚ C_e(45) [i] based on
    1. linear OA(4²¹¹, 4096, F₄, 47) (dual of [4096, 3885, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
    2. linear OA(4²⁰⁵, 4096, F₄, 46) (dual of [4096, 3891, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [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]

(168, 168+47, 1256331)-Net in Base 4 — Upper bound on s

There is no (168, 215, 1256332)-net in base 4, because

1 times m-reduction [i] would yield (168, 214, 1256332)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 693 177637 922911 153258 977951 243416 478261 783973 515594 783505 507962 923051 277759 954641 317861 608833 886639 919036 033607 442585 550383 889072 > 4²¹⁴ [i]