Best Known (33−13, 33, s)-Nets in Base 16
(33−13, 33, 538)-Net over F16 — Constructive and digital
Digital (20, 33, 538)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- 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]
- 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, 13, 257)-net over F256, using
- digital (1, 7, 24)-net over F16, using
(33−13, 33, 743)-Net over F16 — Digital
Digital (20, 33, 743)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1633, 743, F16, 13) (dual of [743, 710, 14]-code), using
- 222 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 10 times 0, 1, 27 times 0, 1, 62 times 0, 1, 116 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 Reed–Solomon code RSe(244,256) [i]
- the expurgated narrow-sense BCH-code C(I) with length 257 | 2562−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 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)
- 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
- 222 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 10 times 0, 1, 27 times 0, 1, 62 times 0, 1, 116 times 0) [i] based on linear OA(1626, 514, F16, 13) (dual of [514, 488, 14]-code), using
(33−13, 33, 527353)-Net in Base 16 — Upper bound on s
There is no (20, 33, 527354)-net in base 16, because
- 1 times m-reduction [i] would yield (20, 32, 527354)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 340 285038 259236 646949 613301 542207 686236 > 1632 [i]