MinT - Best Known (83−28, 83, s)-Nets in Base 16

Best Known (83−28, 83, s)-Nets in Base 16

(83−28, 83, 771)-Net over F₁₆ — Constructive and digital

Digital (55, 83, 771)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (13, 27, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(13,256) in PG(26,16)) for nets [i] based on digital (0, 14, 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 (28, 56, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 28, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(83−28, 83, 4147)-Net over F₁₆ — Digital

Digital (55, 83, 4147)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸³, 4147, F₁₆, 28) (dual of [4147, 4064, 29]-code), using
- 44 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 7 times 0, 1, 33 times 0) [i] based on linear OA(16⁷⁹, 4099, F₁₆, 28) (dual of [4099, 4020, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(26) [i] based on
    1. linear OA(16⁷⁹, 4096, F₁₆, 28) (dual of [4096, 4017, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(16⁷⁶, 4096, F₁₆, 27) (dual of [4096, 4020, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [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]

(83−28, 83, 5547243)-Net in Base 16 — Upper bound on s

There is no (55, 83, 5547244)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8749 015569 382614 623244 608092 515577 187137 492007 970608 506809 433208 410905 448889 927977 604233 294429 614616 > 16⁸³ [i]