MinT - Best Known (49, 49+13, s)-Nets in Base 4

Best Known (49, 49+13, s)-Nets in Base 4

(49, 49+13, 1054)-Net over F₄ — Constructive and digital

Digital (49, 62, 1054)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 26)-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]

(49, 49+13, 3561)-Net over F₄ — Digital

Digital (49, 62, 3561)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶², 3561, F₄, 13) (dual of [3561, 3499, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶², 4105, F₄, 13) (dual of [4105, 4043, 14]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4⁷, 9, F₄, 6) (dual of [9, 2, 7]-code), using
      - 1 times truncation [i] based on linear OA(4⁸, 10, F₄, 7) (dual of [10, 2, 8]-code), using
        repeating each code word 2 times [i] based on linear OA(4³, 5, F₄, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
        extended Reedâ€“Solomon code RS_e(2,4) [i]
        Simplex code S(2,4) [i]
    2. linear OA(4⁵⁵, 4096, F₄, 13) (dual of [4096, 4041, 14]-code), using
      - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(49, 49+13, 1318386)-Net in Base 4 — Upper bound on s

There is no (49, 62, 1318387)-net in base 4, because

1 times m-reduction [i] would yield (49, 61, 1318387)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 316935 574884 074813 206716 306513 933980 > 4⁶¹ [i]