MinT - Best Known (48, 48+13, s)-Nets in Base 4

Best Known (48, 48+13, s)-Nets in Base 4

(48, 48+13, 1046)-Net over F₄ — Constructive and digital

Digital (48, 61, 1046)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 18)-net over F₄, using
  - linear code embedding found in a computer search [i]
2. digital (39, 52, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 13, 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]

(48, 48+13, 3139)-Net over F₄ — Digital

Digital (48, 61, 3139)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶¹, 3139, F₄, 13) (dual of [3139, 3078, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶¹, 4097, F₄, 13) (dual of [4097, 4036, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(48, 48+13, 1046402)-Net in Base 4 — Upper bound on s

There is no (48, 61, 1046403)-net in base 4, because

1 times m-reduction [i] would yield (48, 60, 1046403)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 329229 877048 035038 288013 066482 250720 > 4⁶⁰ [i]