MinT - Best Known (34−13, 34, s)-Nets in Base 16

Best Known (34−13, 34, s)-Nets in Base 16

(34−13, 34, 547)-Net over F₁₆ — Constructive and digital

Digital (21, 34, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 8, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (13, 26, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 13, 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]

(34−13, 34, 919)-Net over F₁₆ — Digital

Digital (21, 34, 919)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16³⁴, 919, F₁₆, 13) (dual of [919, 885, 14]-code), using
- 397 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 10 times 0, 1, 27 times 0, 1, 62 times 0, 1, 116 times 0, 1, 174 times 0) [i] based on linear OA(16²⁶, 514, F₁₆, 13) (dual of [514, 488, 14]-code), using
  - trace code [i] based on linear OA(256¹³, 257, F₂₅₆, 13) (dual of [257, 244, 14]-code or 257-arc in PG(12,256)), using
    - extended Reedâ€“Solomon code RS_e(244,256) [i]
    - the expurgated narrow-sense BCH-code C(I) with length 257Â | 256²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    - algebraic-geometric code AG(F, Q+120P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using the rational function field F₂₅₆(x) [i]
    - algebraic-geometric code AG(F,81P) with degPÂ =Â 3 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)
    - algebraic-geometric code AG(F, Q+48P) with degQ = 3 and degPÂ =Â 5 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)

(34−13, 34, 837122)-Net in Base 16 — Upper bound on s

There is no (21, 34, 837123)-net in base 16, because

1 times m-reduction [i] would yield (21, 33, 837123)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5444 521674 216337 618820 884345 332842 125696 > 16³³ [i]