MinT - Best Known (97, 124, s)-Nets in Base 4

Best Known (97, 124, s)-Nets in Base 4

(97, 124, 1044)-Net over F₄ — Constructive and digital

Digital (97, 124, 1044)-net over F₄, using

trace code for nets [i] based on digital (4, 31, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(97, 124, 3090)-Net over F₄ — Digital

Digital (97, 124, 3090)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁴, 3090, F₄, 27) (dual of [3090, 2966, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²⁴, 4112, F₄, 27) (dual of [4112, 3988, 28]-code), using
  - 2 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]

(97, 124, 939061)-Net in Base 4 — Upper bound on s

There is no (97, 124, 939062)-net in base 4, because

1 times m-reduction [i] would yield (97, 123, 939062)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 113 079424 244212 126682 572726 043354 952494 996488 483883 900849 961896 930452 643914 > 4¹²³ [i]