Best Known (54, 54+28, s)-Nets in Base 16
(54, 54+28, 585)-Net over F16 — Constructive and digital
Digital (54, 82, 585)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 26, 71)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (3, 17, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (2, 9, 33)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (28, 56, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 28, 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, 28, 257)-net over F256, using
- digital (12, 26, 71)-net over F16, using
(54, 54+28, 643)-Net in Base 16 — Constructive
(54, 82, 643)-net in base 16, using
- 1 times m-reduction [i] based on (54, 83, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (11, 25, 129)-net in base 16, using
- base change [i] based on (6, 20, 129)-net in base 32, using
- 1 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
- base change [i] based on digital (0, 15, 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, 15, 129)-net over F128, using
- 1 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
- base change [i] based on (6, 20, 129)-net in base 32, using
- digital (29, 58, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 29, 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, 29, 257)-net over F256, using
- (11, 25, 129)-net in base 16, using
- (u, u+v)-construction [i] based on
(54, 54+28, 3954)-Net over F16 — Digital
Digital (54, 82, 3954)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1682, 3954, F16, 28) (dual of [3954, 3872, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 4111, F16, 28) (dual of [4111, 4029, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- linear OA(1679, 4096, F16, 28) (dual of [4096, 4017, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(1667, 4096, F16, 24) (dual of [4096, 4029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(163, 15, F16, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,16) or 15-cap in PG(2,16)), using
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- Reed–Solomon code RS(13,16) [i]
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(1682, 4111, F16, 28) (dual of [4111, 4029, 29]-code), using
(54, 54+28, 4550598)-Net in Base 16 — Upper bound on s
There is no (54, 82, 4550599)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 546 813395 541618 488184 038107 625955 948827 345479 299305 019254 665715 758778 656002 301256 836583 910115 211416 > 1682 [i]