MinT - Best Known (67−23, 67, s)-Nets in Base 16

Best Known (67−23, 67, s)-Nets in Base 16

(67−23, 67, 771)-Net over F₁₆ — Constructive and digital

Digital (44, 67, 771)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (10, 21, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(10,256) in PG(20,16)) for nets [i] based on digital (0, 11, 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 (23, 46, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 23, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(67−23, 67, 3511)-Net over F₁₆ — Digital

Digital (44, 67, 3511)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁷, 3511, F₁₆, 23) (dual of [3511, 3444, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁶⁷, 4097, F₁₆, 23) (dual of [4097, 4030, 24]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(67−23, 67, 5490884)-Net in Base 16 — Upper bound on s

There is no (44, 67, 5490885)-net in base 16, because

1 times m-reduction [i] would yield (44, 66, 5490885)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 642781 176312 407825 132600 290590 901431 166089 547733 112017 498008 312887 512222 679776 > 16⁶⁶ [i]