MinT - Best Known (58−12, 58, s)-Nets in Base 4

Best Known (58−12, 58, s)-Nets in Base 4

(58−12, 58, 1054)-Net over F₄ — Constructive and digital

Digital (46, 58, 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 (36, 48, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 12, 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]

(58−12, 58, 4072)-Net over F₄ — Digital

Digital (46, 58, 4072)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁸, 4072, F₄, 12) (dual of [4072, 4014, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵⁸, 4117, F₄, 12) (dual of [4117, 4059, 13]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(8) [i] based on
    1. 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]
    2. linear OA(4³⁷, 4096, F₄, 9) (dual of [4096, 4059, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
      - Hamming code H(3,4) [i]

(58−12, 58, 659190)-Net in Base 4 — Upper bound on s

There is no (46, 58, 659191)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 83076 929308 760998 407542 852040 079237 > 4⁵⁸ [i]