MinT - Best Known (223−42, 223, s)-Nets in Base 4

Best Known (223−42, 223, s)-Nets in Base 4

(223−42, 223, 1539)-Net over F₄ — Constructive and digital

Digital (181, 223, 1539)-net over F₄, using

t-expansion [i] based on digital (180, 223, 1539)-net over F₄, using
- 5 times m-reduction [i] based on digital (180, 228, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 76, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(223−42, 223, 11505)-Net over F₄ — Digital

Digital (181, 223, 11505)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²³, 11505, F₄, 42) (dual of [11505, 11282, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4²²³, 16410, F₄, 42) (dual of [16410, 16187, 43]-code), using
  - constructionÂ X applied to C_e(41) âŠ‚ C_e(37) [i] based on
    1. linear OA(4²¹⁸, 16384, F₄, 42) (dual of [16384, 16166, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    2. linear OA(4¹⁹⁷, 16384, F₄, 38) (dual of [16384, 16187, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    3. linear OA(4⁵, 26, F₄, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(223−42, 223, 7156060)-Net in Base 4 — Upper bound on s

There is no (181, 223, 7156061)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 709994 918837 452413 132442 525953 508464 265444 990372 378013 376783 172471 204636 974171 018514 420912 587996 025763 418055 336845 021780 195816 169404 > 4²²³ [i]