MinT - Best Known (226−43, 226, s)-Nets in Base 4

Best Known (226−43, 226, s)-Nets in Base 4

(226−43, 226, 1539)-Net over F₄ — Constructive and digital

Digital (183, 226, 1539)-net over F₄, using

t-expansion [i] based on digital (182, 226, 1539)-net over F₄, using
- 5 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]

(226−43, 226, 10801)-Net over F₄ — Digital

Digital (183, 226, 10801)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²⁶, 10801, F₄, 43) (dual of [10801, 10575, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4²²⁶, 16400, F₄, 43) (dual of [16400, 16174, 44]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,20]) [i] based on
    1. linear OA(4²²⁵, 16385, F₄, 43) (dual of [16385, 16160, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(4²¹¹, 16385, F₄, 41) (dual of [16385, 16174, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
    3. linear OA(4¹, 15, F₄, 1) (dual of [15, 14, 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]

(226−43, 226, 8166071)-Net in Base 4 — Upper bound on s

There is no (183, 226, 8166072)-net in base 4, because

1 times m-reduction [i] would yield (183, 225, 8166072)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2907 359415 846424 411156 076023 255793 426641 306818 067483 614055 353216 979488 799417 790824 128817 294798 836294 771186 843156 169342 192684 175431 053517 > 4²²⁵ [i]