Best Known (65, 65+34, s)-Nets in Base 16
(65, 65+34, 596)-Net over F16 — Constructive and digital
Digital (65, 99, 596)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (14, 31, 82)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (6, 23, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (0, 8, 17)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (34, 68, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 34, 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, 34, 257)-net over F256, using
- digital (14, 31, 82)-net over F16, using
(65, 65+34, 643)-Net in Base 16 — Constructive
(65, 99, 643)-net in base 16, using
- 1 times m-reduction [i] based on (65, 100, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (13, 30, 129)-net in base 16, using
- base change [i] based on (3, 20, 129)-net in base 64, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on (3, 20, 129)-net in base 64, using
- digital (35, 70, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 35, 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, 35, 257)-net over F256, using
- (13, 30, 129)-net in base 16, using
- (u, u+v)-construction [i] based on
(65, 65+34, 4137)-Net over F16 — Digital
Digital (65, 99, 4137)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1699, 4137, F16, 34) (dual of [4137, 4038, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(1699, 4150, F16, 34) (dual of [4150, 4051, 35]-code), using
- 46 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 30 times 0) [i] based on linear OA(1694, 4099, F16, 34) (dual of [4099, 4005, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(1694, 4096, F16, 34) (dual of [4096, 4002, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(1691, 4096, F16, 33) (dual of [4096, 4005, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 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
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 46 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 30 times 0) [i] based on linear OA(1694, 4099, F16, 34) (dual of [4099, 4005, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(1699, 4150, F16, 34) (dual of [4150, 4051, 35]-code), using
(65, 65+34, 4921316)-Net in Base 16 — Upper bound on s
There is no (65, 99, 4921317)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 161391 024312 365541 011881 700759 068828 470130 128490 591302 430226 707039 856563 847789 370956 559439 602360 763528 090501 337407 110136 > 1699 [i]