MinT - Best Known (169, 169+49, s)-Nets in Base 4

Best Known (169, 169+49, s)-Nets in Base 4

(169, 169+49, 1048)-Net over F₄ — Constructive and digital

Digital (169, 218, 1048)-net over F₄, using

4² times duplication [i] based on digital (167, 216, 1048)-net over F₄, using
- trace code for nets [i] based on digital (5, 54, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(169, 169+49, 3650)-Net over F₄ — Digital

Digital (169, 218, 3650)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁸, 3650, F₄, 49) (dual of [3650, 3432, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4²¹⁸, 4103, F₄, 49) (dual of [4103, 3885, 50]-code), using
  - constructionÂ X applied to C_e(48) âŠ‚ C_e(46) [i] based on
    1. linear OA(4²¹⁷, 4096, F₄, 49) (dual of [4096, 3879, 50]-code), using an extension C_e(48) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [i]
    2. 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]
    3. linear OA(4¹, 7, F₄, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(169, 169+49, 907523)-Net in Base 4 — Upper bound on s

There is no (169, 218, 907524)-net in base 4, because

1 times m-reduction [i] would yield (169, 217, 907524)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44363 239972 888503 218090 311039 565739 245279 067832 088631 008519 814549 926450 248802 825610 914053 006012 615613 937736 651521 552789 385640 913184 > 4²¹⁷ [i]