MinT - Best Known (145, 184, s)-Nets in Base 4

Best Known (145, 184, s)-Nets in Base 4

(145, 184, 1056)-Net over F₄ — Constructive and digital

Digital (145, 184, 1056)-net over F₄, using

trace code for nets [i] based on digital (7, 46, 264)-net over F₂₅₆, using
- net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(145, 184, 4265)-Net over F₄ — Digital

Digital (145, 184, 4265)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁴, 4265, F₄, 39) (dual of [4265, 4081, 40]-code), using
- 154 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) [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]

(145, 184, 1662947)-Net in Base 4 — Upper bound on s

There is no (145, 184, 1662948)-net in base 4, because

1 times m-reduction [i] would yield (145, 183, 1662948)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 150 307280 082126 858286 748188 574530 137407 543039 573004 634338 825940 578418 265302 356068 135622 343612 474794 434431 698886 > 4¹⁸³ [i]