Best Known (7, 7+6, s)-Nets in Base 16
(7, 7+6, 514)-Net over F16 — Constructive and digital
Digital (7, 13, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (7, 14, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 7, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction 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]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 7, 257)-net over F256, using
(7, 7+6, 516)-Net over F16 — Digital
Digital (7, 13, 516)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1613, 516, F16, 6) (dual of [516, 503, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1612, 514, F16, 6) (dual of [514, 502, 7]-code), using
- trace code [i] based on linear OA(2566, 257, F256, 6) (dual of [257, 251, 7]-code or 257-arc in PG(5,256)), using
- extended Reed–Solomon code RSe(251,256) [i]
- algebraic-geometric code AG(F,125P) 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 = 4 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,50P) with degPÂ =Â 5 [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257 (see above)
- trace code [i] based on linear OA(2566, 257, F256, 6) (dual of [257, 251, 7]-code or 257-arc in PG(5,256)), using
- linear OA(1612, 515, F16, 5) (dual of [515, 503, 6]-code), using Gilbert–Varšamov bound and bm = 1612 > Vbs−1(k−1) = 145 596792 247936 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 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
- linear OA(1612, 514, F16, 6) (dual of [514, 502, 7]-code), using
- construction X with Varšamov bound [i] based on
(7, 7+6, 20004)-Net in Base 16 — Upper bound on s
There is no (7, 13, 20005)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 4504 186279 028476 > 1613 [i]