MinT - Best Known (172, 219, s)-Nets in Base 4

Best Known (172, 219, s)-Nets in Base 4

(172, 219, 1056)-Net over F₄ — Constructive and digital

Digital (172, 219, 1056)-net over F₄, using

1 times m-reduction [i] based on digital (172, 220, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 55, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(172, 219, 4444)-Net over F₄ — Digital

Digital (172, 219, 4444)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁹, 4444, F₄, 47) (dual of [4444, 4225, 48]-code), using
- 334 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0, 1, 61 times 0, 1, 89 times 0, 1, 111 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]

(172, 219, 1598864)-Net in Base 4 — Upper bound on s

There is no (172, 219, 1598865)-net in base 4, because

1 times m-reduction [i] would yield (172, 218, 1598865)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177452 592290 494866 098803 821831 238891 700555 095880 258251 161057 449511 554563 202397 096738 932575 300521 879101 909030 912658 022883 833507 332736 > 4²¹⁸ [i]