MinT - Best Known (186−39, 186, s)-Nets in Base 4

Best Known (186−39, 186, s)-Nets in Base 4

(186−39, 186, 1056)-Net over F₄ — Constructive and digital

Digital (147, 186, 1056)-net over F₄, using

4² times duplication [i] based on 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]

(186−39, 186, 4484)-Net over F₄ — Digital

Digital (147, 186, 4484)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁶, 4484, F₄, 39) (dual of [4484, 4298, 40]-code), using
- 371 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) [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]

(186−39, 186, 1924216)-Net in Base 4 — Upper bound on s

There is no (147, 186, 1924217)-net in base 4, because

1 times m-reduction [i] would yield (147, 185, 1924217)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2404 926837 661876 848320 261141 104759 624932 306789 415114 582911 292349 264255 962692 560340 315959 622486 655971 763686 541048 > 4¹⁸⁵ [i]