MinT - Best Known (59, 74, s)-Nets in Base 4

Best Known (59, 74, s)-Nets in Base 4

(59, 74, 1062)-Net over F₄ — Constructive and digital

Digital (59, 74, 1062)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 14, 34)-net over F₄, using
  - trace code for nets [i] based on digital (0, 7, 17)-net over F₁₆, using
    - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
      - Niederreiter sequence [i]
2. digital (45, 60, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 15, 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]

(59, 74, 4164)-Net over F₄ — Digital

Digital (59, 74, 4164)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁴, 4164, F₄, 15) (dual of [4164, 4090, 16]-code), using
- 49 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0) [i] based on linear OA(4⁶⁸, 4109, F₄, 15) (dual of [4109, 4041, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(12) [i] based on
    1. linear OA(4⁶⁷, 4096, F₄, 15) (dual of [4096, 4029, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [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]
    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]

(59, 74, 2140040)-Net in Base 4 — Upper bound on s

There is no (59, 74, 2140041)-net in base 4, because

1 times m-reduction [i] would yield (59, 73, 2140041)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 89 203209 405597 240380 611233 142617 634838 065768 > 4⁷³ [i]