MinT - Best Known (165, 211, s)-Nets in Base 4

Best Known (165, 211, s)-Nets in Base 4

(165, 211, 1052)-Net over F₄ — Constructive and digital

Digital (165, 211, 1052)-net over F₄, using

1 times m-reduction [i] based on digital (165, 212, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 53, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(165, 211, 4148)-Net over F₄ — Digital

Digital (165, 211, 4148)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹¹, 4148, F₄, 46) (dual of [4148, 3937, 47]-code), using
- 40 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 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]

(165, 211, 1048511)-Net in Base 4 — Upper bound on s

There is no (165, 211, 1048512)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 830801 040094 876745 675035 381531 584972 111130 238041 051220 676152 541979 350628 946977 552000 861053 170176 332348 752538 677708 534288 679933 > 4²¹¹ [i]