MinT - Best Known (57, 76, s)-Nets in Base 4

Best Known (57, 76, s)-Nets in Base 4

(57, 76, 1028)-Net over F₄ — Constructive and digital

Digital (57, 76, 1028)-net over F₄, using

trace code for nets [i] based on digital (0, 19, 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]

(57, 76, 1054)-Net over F₄ — Digital

Digital (57, 76, 1054)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁶, 1054, F₄, 19) (dual of [1054, 978, 20]-code), using
- 14 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 0, 0, 0, 1, 7 times 0) [i] based on linear OA(4⁷², 1036, F₄, 19) (dual of [1036, 964, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
    1. linear OA(4⁷¹, 1025, F₄, 19) (dual of [1025, 954, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(4⁶¹, 1025, F₄, 17) (dual of [1025, 964, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
    3. linear OA(4¹, 11, F₄, 1) (dual of [11, 10, 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]

(57, 76, 143805)-Net in Base 4 — Upper bound on s

There is no (57, 76, 143806)-net in base 4, because

1 times m-reduction [i] would yield (57, 75, 143806)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1427 287852 046309 385456 364592 145829 729424 167927 > 4⁷⁵ [i]