MinT - Best Known (222−47, 222, s)-Nets in Base 4

Best Known (222−47, 222, s)-Nets in Base 4

(222−47, 222, 1060)-Net over F₄ — Constructive and digital

Digital (175, 222, 1060)-net over F₄, using

4² times duplication [i] based on digital (173, 220, 1060)-net over F₄, using
- trace code for nets [i] based on digital (8, 55, 265)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(222−47, 222, 4851)-Net over F₄ — Digital

Digital (175, 222, 4851)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²², 4851, F₄, 47) (dual of [4851, 4629, 48]-code), using
- 738 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0, 1, 61 times 0, 1, 89 times 0, 1, 111 times 0, 1, 126 times 0, 1, 134 times 0, 1, 141 times 0) [i] based on linear OA(4²¹¹, 4102, F₄, 47) (dual of [4102, 3891, 48]-code), using
  - constructionÂ X applied to C_e(46) âŠ‚ C_e(45) [i] based on
    1. linear OA(4²¹¹, 4096, F₄, 47) (dual of [4096, 3885, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
    2. 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]
    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]

(222−47, 222, 1915764)-Net in Base 4 — Upper bound on s

There is no (175, 222, 1915765)-net in base 4, because

1 times m-reduction [i] would yield (175, 221, 1915765)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 356987 672498 585599 433095 825567 808995 614006 408255 751131 018878 708875 607525 029876 000432 856465 631760 882058 615076 630467 186345 612096 426336 > 4²²¹ [i]