MinT - Best Known (171, 171+46, s)-Nets in Base 4

Best Known (171, 171+46, s)-Nets in Base 4

(171, 171+46, 1060)-Net over F₄ — Constructive and digital

Digital (171, 217, 1060)-net over F₄, using

4¹ times duplication [i] based on digital (170, 216, 1060)-net over F₄, using
- trace code for nets [i] based on digital (8, 54, 265)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(171, 171+46, 4726)-Net over F₄ — Digital

Digital (171, 217, 4726)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁷, 4726, F₄, 46) (dual of [4726, 4509, 47]-code), using
- 612 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 37 times 0, 1, 61 times 0, 1, 89 times 0, 1, 113 times 0, 1, 128 times 0, 1, 138 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]

(171, 171+46, 1505340)-Net in Base 4 — Upper bound on s

There is no (171, 217, 1505341)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44362 986133 424241 279313 695600 876679 306032 310187 132160 477791 671537 986384 447124 974221 744803 711680 722062 780363 687710 620240 215582 737976 > 4²¹⁷ [i]