MinT - Best Known (61, 61+32, s)-Nets in Base 16

Best Known (61, 61+32, s)-Nets in Base 16

(61, 61+32, 585)-Net over F₁₆ — Constructive and digital

Digital (61, 93, 585)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (13, 29, 71)-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 (3, 19, 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 (32, 64, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 32, 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]

(61, 61+32, 643)-Net in Base 16 — Constructive

(61, 93, 643)-net in base 16, using

1 times m-reduction [i] based on (61, 94, 643)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (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 (33, 66, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 33, 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]

(61, 61+32, 3941)-Net over F₁₆ — Digital

Digital (61, 93, 3941)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹³, 3941, F₁₆, 32) (dual of [3941, 3848, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹³, 4107, F₁₆, 32) (dual of [4107, 4014, 33]-code), using
  - 1 times truncation [i] based on linear OA(16⁹⁴, 4108, F₁₆, 33) (dual of [4108, 4014, 34]-code), using
    - constructionÂ X applied to C_e(32) âŠ‚ C_e(28) [i] based on
      1. linear OA(16⁹¹, 4096, F₁₆, 33) (dual of [4096, 4005, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [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(16³, 12, F₁₆, 3) (dual of [12, 9, 4]-code or 12-arc in PG(2,16) or 12-cap in PG(2,16)), using
        discarding factors / shortening the dual code based on linear OA(16³, 16, F₁₆, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
        Reedâ€“Solomon code RS(13,16) [i]

(61, 61+32, 4522661)-Net in Base 16 — Upper bound on s

There is no (61, 93, 4522662)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9619 662297 927226 394531 136065 346797 940769 436853 684409 139044 994107 466822 469096 427312 250034 184659 674522 124261 808006 > 16⁹³ [i]