MinT - Best Known (89−30, 89, s)-Nets in Base 16

Best Known (89−30, 89, s)-Nets in Base 16

(89−30, 89, 771)-Net over F₁₆ — Constructive and digital

Digital (59, 89, 771)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (14, 29, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(14,256) in PG(28,16)) for nets [i] based on digital (0, 15, 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 (30, 60, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 30, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(89−30, 89, 4180)-Net over F₁₆ — Digital

Digital (59, 89, 4180)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁹, 4180, F₁₆, 30) (dual of [4180, 4091, 31]-code), using
- 77 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 14 times 0, 1, 57 times 0) [i] based on linear OA(16⁸⁵, 4099, F₁₆, 30) (dual of [4099, 4014, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(16⁸⁵, 4096, F₁₆, 30) (dual of [4096, 4011, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [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(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]

(89−30, 89, 5972003)-Net in Base 16 — Upper bound on s

There is no (59, 89, 5972004)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146784 213591 428213 333535 790048 003368 770898 742636 429804 526349 299215 890237 984377 595412 875158 542894 512059 552776 > 16⁸⁹ [i]