MinT - Best Known (60, 60+16, s)-Nets in Base 4

Best Known (60, 60+16, s)-Nets in Base 4

(60, 60+16, 1045)-Net over F₄ — Constructive and digital

Digital (60, 76, 1045)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 17)-net over F₄, using
  - linear code embedding found in a computer search [i]
2. digital (48, 64, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 16, 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]

(60, 60+16, 3375)-Net over F₄ — Digital

Digital (60, 76, 3375)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁶, 3375, F₄, 16) (dual of [3375, 3299, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷⁶, 4117, F₄, 16) (dual of [4117, 4041, 17]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(12) [i] based on
    1. linear OA(4⁷³, 4096, F₄, 17) (dual of [4096, 4023, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
      - Hamming code H(3,4) [i]

(60, 60+16, 657861)-Net in Base 4 — Upper bound on s

There is no (60, 76, 657862)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5709 005877 976688 014055 779024 737811 740664 017880 > 4⁷⁶ [i]