Best Known (57, 57+32, s)-Nets in Base 16
(57, 57+32, 581)-Net over F16 — Constructive and digital
Digital (57, 89, 581)-net over F16, using
- 1 times m-reduction [i] based on digital (57, 90, 581)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 22, 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, 68, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 34, 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, 34, 258)-net over F256, using
- digital (6, 22, 65)-net over F16, using
- (u, u+v)-construction [i] based on
(57, 57+32, 2387)-Net over F16 — Digital
Digital (57, 89, 2387)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1689, 2387, F16, 32) (dual of [2387, 2298, 33]-code), using
- 2297 step Varšamov–Edel lengthening with (ri) = (3, 2, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 17 times 0, 1, 17 times 0, 1, 20 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 29 times 0, 1, 32 times 0, 1, 35 times 0, 1, 39 times 0, 1, 42 times 0, 1, 46 times 0, 1, 51 times 0, 1, 56 times 0, 1, 62 times 0, 1, 67 times 0, 1, 74 times 0, 1, 81 times 0, 1, 89 times 0, 1, 97 times 0, 1, 106 times 0, 1, 117 times 0, 1, 128 times 0, 1, 139 times 0, 1, 153 times 0, 1, 168 times 0, 1, 184 times 0, 1, 201 times 0) [i] based on linear OA(1632, 33, F16, 32) (dual of [33, 1, 33]-code or 33-arc in PG(31,16)), using
- dual of repetition code with length 33 [i]
- 2297 step Varšamov–Edel lengthening with (ri) = (3, 2, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 17 times 0, 1, 17 times 0, 1, 20 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 29 times 0, 1, 32 times 0, 1, 35 times 0, 1, 39 times 0, 1, 42 times 0, 1, 46 times 0, 1, 51 times 0, 1, 56 times 0, 1, 62 times 0, 1, 67 times 0, 1, 74 times 0, 1, 81 times 0, 1, 89 times 0, 1, 97 times 0, 1, 106 times 0, 1, 117 times 0, 1, 128 times 0, 1, 139 times 0, 1, 153 times 0, 1, 168 times 0, 1, 184 times 0, 1, 201 times 0) [i] based on linear OA(1632, 33, F16, 32) (dual of [33, 1, 33]-code or 33-arc in PG(31,16)), using
(57, 57+32, 2261326)-Net in Base 16 — Upper bound on s
There is no (57, 89, 2261327)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 146784 692111 154131 572319 986268 354427 048519 699032 552745 814496 319527 526072 223277 607641 581996 430039 102104 106356 > 1689 [i]