MinT - Best Known (182, 225, s)-Nets in Base 4

Best Known (182, 225, s)-Nets in Base 4

(182, 225, 1539)-Net over F₄ — Constructive and digital

Digital (182, 225, 1539)-net over F₄, using

6 times m-reduction [i] based on digital (182, 231, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 77, 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]

(182, 225, 10441)-Net over F₄ — Digital

Digital (182, 225, 10441)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²⁵, 10441, F₄, 43) (dual of [10441, 10216, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4²²⁵, 16384, F₄, 43) (dual of [16384, 16159, 44]-code), using
  - an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]

(182, 225, 7644403)-Net in Base 4 — Upper bound on s

There is no (182, 225, 7644404)-net in base 4, because

1 times m-reduction [i] would yield (182, 224, 7644404)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 726 840254 586119 635788 189339 989936 133335 104612 350036 587098 877508 257464 875744 046031 958582 102448 412595 895513 633842 490561 572773 715400 480648 > 4²²⁴ [i]