MinT - Best Known (182−41, 182, s)-Nets in Base 4

Best Known (182−41, 182, s)-Nets in Base 4

(182−41, 182, 1044)-Net over F₄ — Constructive and digital

Digital (141, 182, 1044)-net over F₄, using

4² times duplication [i] based on digital (139, 180, 1044)-net over F₄, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(182−41, 182, 3164)-Net over F₄ — Digital

Digital (141, 182, 3164)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸², 3164, F₄, 41) (dual of [3164, 2982, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁸², 4103, F₄, 41) (dual of [4103, 3921, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(38) [i] based on
    1. linear OA(4¹⁸¹, 4096, F₄, 41) (dual of [4096, 3915, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(4¹⁷⁵, 4096, F₄, 39) (dual of [4096, 3921, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [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]

(182−41, 182, 777712)-Net in Base 4 — Upper bound on s

There is no (141, 182, 777713)-net in base 4, because

1 times m-reduction [i] would yield (141, 181, 777713)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 394381 713295 982320 461066 230356 126026 514687 413411 717030 256042 193171 809439 750538 946999 248873 461283 399406 404856 > 4¹⁸¹ [i]