Best Known (86−30, 86, s)-Nets in Base 16
(86−30, 86, 583)-Net over F16 — Constructive and digital
Digital (56, 86, 583)-net over F16, using
- 1 times m-reduction [i] based on digital (56, 87, 583)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 21, 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 (35, 66, 518)-net over F16, using
- trace code for nets [i] based on digital (2, 33, 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, 33, 259)-net over F256, using
- digital (6, 21, 65)-net over F16, using
- (u, u+v)-construction [i] based on
(86−30, 86, 596)-Net in Base 16 — Constructive
(56, 86, 596)-net in base 16, using
- (u, u+v)-construction [i] based on
- (9, 24, 80)-net in base 16, using
- base change [i] based on digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- base change [i] based on digital (1, 16, 80)-net over F64, using
- digital (32, 62, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 31, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 31, 258)-net over F256, using
- (9, 24, 80)-net in base 16, using
(86−30, 86, 3392)-Net over F16 — Digital
Digital (56, 86, 3392)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1686, 3392, F16, 30) (dual of [3392, 3306, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(1686, 4103, F16, 30) (dual of [4103, 4017, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- linear OA(1685, 4096, F16, 30) (dual of [4096, 4011, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- 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(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(29) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(1686, 4103, F16, 30) (dual of [4103, 4017, 31]-code), using
(86−30, 86, 3430011)-Net in Base 16 — Upper bound on s
There is no (56, 86, 3430012)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 35 835961 403808 236041 842118 159189 723947 688105 301127 546201 197268 517427 447032 697362 265694 114214 696234 491076 > 1686 [i]