MinT - Best Known (58−16, 58, s)-Nets in Base 8

Best Known (58−16, 58, s)-Nets in Base 8

(58−16, 58, 513)-Net over F₈ — Constructive and digital

Digital (42, 58, 513)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵⁸, 513, F₈, 16, 16) (dual of [(513, 16), 8150, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8⁵⁸, 4104, F₈, 16) (dual of [4104, 4046, 17]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁵⁸, 4105, F₈, 16) (dual of [4105, 4047, 17]-code), using
    - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [i] based on
      1. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      2. linear OA(8⁴⁹, 4096, F₈, 14) (dual of [4096, 4047, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
      3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(58−16, 58, 547)-Net in Base 8 — Constructive

(42, 58, 547)-net in base 8, using

8² times duplication [i] based on (40, 56, 547)-net in base 8, using
- base change [i] based on digital (26, 42, 547)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 10, 33)-net over F₁₆, using
      - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
    2. digital (16, 32, 514)-net over F₁₆, 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]

(58−16, 58, 4096)-Net over F₈ — Digital

Digital (42, 58, 4096)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁸, 4096, F₈, 16) (dual of [4096, 4038, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵⁸, 4105, F₈, 16) (dual of [4105, 4047, 17]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [i] based on
    1. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(8⁴⁹, 4096, F₈, 14) (dual of [4096, 4047, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(58−16, 58, 1896676)-Net in Base 8 — Upper bound on s

There is no (42, 58, 1896677)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 23945 247134 798944 093107 159426 227318 123361 359002 711148 > 8⁵⁸ [i]