MinT - Best Known (89−31, 89, s)-Nets in Base 16

Best Known (89−31, 89, s)-Nets in Base 16

(89−31, 89, 585)-Net over F₁₆ — Constructive and digital

Digital (58, 89, 585)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (12, 27, 71)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 9, 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 (3, 18, 38)-net over F₁₆, using
      - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (31, 62, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 31, 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]

(89−31, 89, 643)-Net in Base 16 — Constructive

(58, 89, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (12, 27, 129)-net in base 16, using
  - 1 times m-reduction [i] based on (12, 28, 129)-net in base 16, using
    - base change [i] based on digital (0, 16, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. digital (31, 62, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 31, 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]

(89−31, 89, 3492)-Net over F₁₆ — Digital

Digital (58, 89, 3492)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁹, 3492, F₁₆, 31) (dual of [3492, 3403, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸⁹, 4103, F₁₆, 31) (dual of [4103, 4014, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(28) [i] based on
    1. linear OA(16⁸⁸, 4096, F₁₆, 31) (dual of [4096, 4008, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(16⁸², 4096, F₁₆, 29) (dual of [4096, 4014, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(16¹, 7, F₁₆, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(89−31, 89, 4964154)-Net in Base 16 — Upper bound on s

There is no (58, 89, 4964155)-net in base 16, because

1 times m-reduction [i] would yield (58, 88, 4964155)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9174 021925 691976 011992 215954 351713 813826 421755 171073 662418 970987 049407 495221 395769 623059 169707 517980 367376 > 16⁸⁸ [i]