MinT - Best Known (66, 100, s)-Nets in Base 16

Best Known (66, 100, s)-Nets in Base 16

(66, 100, 603)-Net over F₁₆ — Constructive and digital

Digital (66, 100, 603)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (15, 32, 89)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 9, 24)-net over F₁₆, using
      - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
    2. digital (6, 23, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
2. digital (34, 68, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 34, 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]

(66, 100, 645)-Net in Base 16 — Constructive

(66, 100, 645)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (13, 30, 129)-net in base 16, using
  - base change [i] based on (3, 20, 129)-net in base 64, using
    - 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
      - base change [i] based on digital (0, 18, 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 (36, 70, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 35, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(66, 100, 4228)-Net over F₁₆ — Digital

Digital (66, 100, 4228)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰⁰, 4228, F₁₆, 34) (dual of [4228, 4128, 35]-code), using
- 123 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 76 times 0) [i] based on linear OA(16⁹⁴, 4099, F₁₆, 34) (dual of [4099, 4005, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
    1. linear OA(16⁹⁴, 4096, F₁₆, 34) (dual of [4096, 4002, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. 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]
    3. linear OA(16⁰, 3, F₁₆, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(66, 100, 5793112)-Net in Base 16 — Upper bound on s

There is no (66, 100, 5793113)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 582251 431948 981122 327988 822932 884752 048779 312415 461348 719293 956653 575528 737198 166121 230916 372854 928029 681820 554692 963416 > 16¹⁰⁰ [i]