MinT - Best Known (123−27, 123, s)-Nets in Base 4

Best Known (123−27, 123, s)-Nets in Base 4

(123−27, 123, 1040)-Net over F₄ — Constructive and digital

Digital (96, 123, 1040)-net over F₄, using

1 times m-reduction [i] based on digital (96, 124, 1040)-net over F₄, using
- trace code for nets [i] based on digital (3, 31, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(123−27, 123, 2922)-Net over F₄ — Digital

Digital (96, 123, 2922)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²³, 2922, F₄, 27) (dual of [2922, 2799, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²³, 4111, F₄, 27) (dual of [4111, 3988, 28]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹²², 4110, F₄, 27) (dual of [4110, 3988, 28]-code), using
    - constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(4¹²¹, 4097, F₄, 27) (dual of [4097, 3976, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
      2. linear OA(4¹⁰⁹, 4097, F₄, 25) (dual of [4097, 3988, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. linear OA(4¹, 13, F₄, 1) (dual of [13, 12, 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]

(123−27, 123, 844075)-Net in Base 4 — Upper bound on s

There is no (96, 123, 844076)-net in base 4, because

1 times m-reduction [i] would yield (96, 122, 844076)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28 269966 043900 062846 736572 771635 231931 553446 942562 440827 851225 260467 097784 > 4¹²² [i]