MinT - Best Known (110−23, 110, s)-Nets in Base 4

Best Known (110−23, 110, s)-Nets in Base 4

(110−23, 110, 1049)-Net over F₄ — Constructive and digital

Digital (87, 110, 1049)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 18, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (69, 92, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 23, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(110−23, 110, 3842)-Net over F₄ — Digital

Digital (87, 110, 3842)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁰, 3842, F₄, 23) (dual of [3842, 3732, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹¹⁰, 4110, F₄, 23) (dual of [4110, 4000, 24]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,10]) [i] based on
    1. 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]
    2. linear OA(4⁹⁷, 4097, F₄, 21) (dual of [4097, 4000, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (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]

(110−23, 110, 1512716)-Net in Base 4 — Upper bound on s

There is no (87, 110, 1512717)-net in base 4, because

1 times m-reduction [i] would yield (87, 109, 1512717)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 421250 099625 157688 802863 153225 732080 303942 036819 474819 725234 282460 > 4¹⁰⁹ [i]