Best Known (61, 61+38, s)-Nets in Base 16
(61, 61+38, 559)-Net over F16 — Constructive and digital
Digital (61, 99, 559)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 23, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, 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
- digital (4, 23, 45)-net over F16, using
(61, 61+38, 1647)-Net over F16 — Digital
Digital (61, 99, 1647)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1699, 1647, F16, 38) (dual of [1647, 1548, 39]-code), using
- 1547 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 5 times 0, 1, 4 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 27 times 0, 1, 28 times 0, 1, 31 times 0, 1, 33 times 0, 1, 37 times 0, 1, 39 times 0, 1, 42 times 0, 1, 46 times 0, 1, 50 times 0, 1, 53 times 0, 1, 58 times 0, 1, 62 times 0, 1, 68 times 0, 1, 73 times 0, 1, 79 times 0, 1, 85 times 0, 1, 92 times 0, 1, 99 times 0, 1, 107 times 0, 1, 116 times 0) [i] based on linear OA(1638, 39, F16, 38) (dual of [39, 1, 39]-code or 39-arc in PG(37,16)), using
- dual of repetition code with length 39 [i]
- 1547 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 5 times 0, 1, 4 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 27 times 0, 1, 28 times 0, 1, 31 times 0, 1, 33 times 0, 1, 37 times 0, 1, 39 times 0, 1, 42 times 0, 1, 46 times 0, 1, 50 times 0, 1, 53 times 0, 1, 58 times 0, 1, 62 times 0, 1, 68 times 0, 1, 73 times 0, 1, 79 times 0, 1, 85 times 0, 1, 92 times 0, 1, 99 times 0, 1, 107 times 0, 1, 116 times 0) [i] based on linear OA(1638, 39, F16, 38) (dual of [39, 1, 39]-code or 39-arc in PG(37,16)), using
(61, 61+38, 993616)-Net in Base 16 — Upper bound on s
There is no (61, 99, 993617)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 161393 416086 342200 539919 736577 196599 819561 127646 842421 779678 699614 064536 762562 821236 034001 465836 356124 887689 681111 960896 > 1699 [i]