MinT - Best Known (21−8, 21, s)-Nets in Base 16

Best Known (21−8, 21, s)-Nets in Base 16

(21−8, 21, 538)-Net over F₁₆ — Constructive and digital

Digital (13, 21, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (8, 16, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 8, 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]

(21−8, 21, 547)-Net in Base 16 — Constructive

(13, 21, 547)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (1, 5, 33)-net in base 16, using
  - base change [i] based on digital (0, 4, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
      - Niederreiter sequence [i]
2. digital (8, 16, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 8, 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]

(21−8, 21, 940)-Net over F₁₆ — Digital

Digital (13, 21, 940)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16²¹, 940, F₁₆, 8) (dual of [940, 919, 9]-code), using
- 421 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 7 times 0, 1, 36 times 0, 1, 115 times 0, 1, 259 times 0) [i] based on linear OA(16¹⁶, 514, F₁₆, 8) (dual of [514, 498, 9]-code), using
  - trace code [i] based on linear OA(256⁸, 257, F₂₅₆, 8) (dual of [257, 249, 9]-code or 257-arc in PG(7,256)), using
    - extended Reedâ€“Solomon code RS_e(249,256) [i]
    - algebraic-geometric code AG(F,124P) with 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, Q+82P) with degQ = 2 and 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+49P) with degQ = 3 and degPÂ =Â 5 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)

(21−8, 21, 309448)-Net in Base 16 — Upper bound on s

There is no (13, 21, 309449)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 19 342831 406240 268284 972566 > 16²¹ [i]