Best Known (83−29, 83, s)-Nets in Base 16
(83−29, 83, 583)-Net over F16 — Constructive and digital
Digital (54, 83, 583)-net over F16, using
- 161 times duplication [i] based on digital (53, 82, 583)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 20, 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 (33, 62, 518)-net over F16, using
- trace code for nets [i] based on digital (2, 31, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 31, 259)-net over F256, using
- digital (6, 20, 65)-net over F16, using
- (u, u+v)-construction [i] based on
(83−29, 83, 643)-Net in Base 16 — Constructive
(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
(83−29, 83, 3292)-Net over F16 — Digital
Digital (54, 83, 3292)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1683, 3292, F16, 29) (dual of [3292, 3209, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(1683, 4103, F16, 29) (dual of [4103, 4020, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(1682, 4096, F16, 29) (dual of [4096, 4014, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(1676, 4096, F16, 27) (dual of [4096, 4020, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(1683, 4103, F16, 29) (dual of [4103, 4020, 30]-code), using
(83−29, 83, 4550598)-Net in Base 16 — Upper bound on s
There is no (54, 83, 4550599)-net in base 16, because
- 1 times m-reduction [i] would yield (54, 82, 4550599)-net in base 16, but
- 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]