MinT - Best Known (167, 167+47, s)-Nets in Base 4

Best Known (167, 167+47, s)-Nets in Base 4

(167, 167+47, 1052)-Net over F₄ — Constructive and digital

Digital (167, 214, 1052)-net over F₄, using

4² times duplication [i] based on digital (165, 212, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 53, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(167, 167+47, 4117)-Net over F₄ — Digital

Digital (167, 214, 4117)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁴, 4117, F₄, 47) (dual of [4117, 3903, 48]-code), using
- 12 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 8 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]

(167, 167+47, 1182843)-Net in Base 4 — Upper bound on s

There is no (167, 214, 1182844)-net in base 4, because

1 times m-reduction [i] would yield (167, 213, 1182844)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 173 294149 597920 352475 235320 965980 445675 756644 209017 279568 952583 293659 031333 122775 910688 618568 689745 035837 341005 470406 770735 314558 > 4²¹³ [i]