MinT - Best Known (189−39, 189, s)-Nets in Base 4

Best Known (189−39, 189, s)-Nets in Base 4

(189−39, 189, 1060)-Net over F₄ — Constructive and digital

Digital (150, 189, 1060)-net over F₄, using

4¹ times duplication [i] based on digital (149, 188, 1060)-net over F₄, using
- trace code for nets [i] based on digital (8, 47, 265)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(189−39, 189, 4966)-Net over F₄ — Digital

Digital (150, 189, 4966)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁹, 4966, F₄, 39) (dual of [4966, 4777, 40]-code), using
- 850 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 40 times 0, 1, 65 times 0, 1, 93 times 0, 1, 122 times 0, 1, 144 times 0, 1, 160 times 0, 1, 172 times 0) [i] based on linear OA(4¹⁷⁵, 4102, F₄, 39) (dual of [4102, 3927, 40]-code), using
  - constructionÂ X applied to C_e(38) âŠ‚ C_e(37) [i] based on
    1. linear OA(4¹⁷⁵, 4096, F₄, 39) (dual of [4096, 3921, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    2. linear OA(4¹⁶⁹, 4096, F₄, 38) (dual of [4096, 3927, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [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]

(189−39, 189, 2395061)-Net in Base 4 — Upper bound on s

There is no (150, 189, 2395062)-net in base 4, because

1 times m-reduction [i] would yield (150, 188, 2395062)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 153914 838548 033791 970755 598973 628361 907635 327936 277813 278911 003602 869373 498506 397453 540531 389149 567135 920994 852690 > 4¹⁸⁸ [i]