Best Known (71, 109, s)-Nets in Base 16
(71, 109, 587)-Net over F16 — Constructive and digital
Digital (71, 109, 587)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 25, 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 (46, 84, 522)-net over F16, using
- trace code for nets [i] based on digital (4, 42, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 42, 261)-net over F256, using
- digital (6, 25, 65)-net over F16, using
(71, 109, 618)-Net in Base 16 — Constructive
(71, 109, 618)-net in base 16, using
- (u, u+v)-construction [i] based on
- (14, 33, 104)-net in base 16, using
- base change [i] based on digital (3, 22, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- base change [i] based on digital (3, 22, 104)-net over F64, using
- digital (38, 76, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 38, 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, 38, 257)-net over F256, using
- (14, 33, 104)-net in base 16, using
(71, 109, 3880)-Net over F16 — Digital
Digital (71, 109, 3880)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16109, 3880, F16, 38) (dual of [3880, 3771, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(16109, 4111, F16, 38) (dual of [4111, 4002, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(16106, 4096, F16, 38) (dual of [4096, 3990, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- 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(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(37) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(16109, 4111, F16, 38) (dual of [4111, 4002, 39]-code), using
(71, 109, 4275327)-Net in Base 16 — Upper bound on s
There is no (71, 109, 4275328)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 177451 003132 570361 563079 879077 187766 436220 319311 290675 850743 375653 356435 247311 917071 793436 860984 049889 585891 879996 416834 158512 223081 > 16109 [i]