MinT - Best Known (61−20, 61, s)-Nets in Base 16

Best Known (61−20, 61, s)-Nets in Base 16

(61−20, 61, 1028)-Net over F₁₆ — Constructive and digital

Digital (41, 61, 1028)-net over F₁₆, using

1 times m-reduction [i] based on digital (41, 62, 1028)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 20, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 10, 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]
  2. digital (21, 42, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 21, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(61−20, 61, 4335)-Net over F₁₆ — Digital

Digital (41, 61, 4335)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶¹, 4335, F₁₆, 20) (dual of [4335, 4274, 21]-code), using
- 230 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 15 times 0, 1, 54 times 0, 1, 154 times 0) [i] based on linear OA(16⁵⁵, 4099, F₁₆, 20) (dual of [4099, 4044, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
    1. linear OA(16⁵⁵, 4096, F₁₆, 20) (dual of [4096, 4041, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(16⁵², 4096, F₁₆, 19) (dual of [4096, 4044, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]

(61−20, 61, 6683694)-Net in Base 16 — Upper bound on s

There is no (41, 61, 6683695)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 28 269566 451728 816040 511355 374450 840010 422306 748117 920678 894605 295614 880501 > 16⁶¹ [i]