MinT - Best Known (164, 164+46, s)-Nets in Base 4

Best Known (164, 164+46, s)-Nets in Base 4

(164, 164+46, 1052)-Net over F₄ — Constructive and digital

Digital (164, 210, 1052)-net over F₄, using

4² times duplication [i] based on digital (162, 208, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 52, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(164, 164+46, 4126)-Net over F₄ — Digital

Digital (164, 210, 4126)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁰, 4126, F₄, 46) (dual of [4126, 3916, 47]-code), using
- 19 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 1, 4 times 0, 1, 10 times 0) [i] based on linear OA(4²⁰⁵, 4102, F₄, 46) (dual of [4102, 3897, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(44) [i] based on
    1. 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]
    2. linear OA(4¹⁹⁹, 4096, F₄, 45) (dual of [4096, 3897, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [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]

(164, 164+46, 987179)-Net in Base 4 — Upper bound on s

There is no (164, 210, 987180)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 707692 309835 203242 182564 322347 209034 351595 021827 010151 379878 193823 643541 290882 046277 816375 340227 491152 469790 631660 696016 876276 > 4²¹⁰ [i]