Best Known (128, 128+33, s)-Nets in Base 4
(128, 128+33, 1058)-Net over F4 — Constructive and digital
Digital (128, 161, 1058)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 29, 30)-net over F4, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- F4 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- digital (99, 132, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 33, 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, 33, 257)-net over F256, using
- digital (13, 29, 30)-net over F4, using
(128, 128+33, 4641)-Net over F4 — Digital
Digital (128, 161, 4641)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4161, 4641, F4, 33) (dual of [4641, 4480, 34]-code), using
- 528 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0, 1, 43 times 0, 1, 62 times 0, 1, 85 times 0, 1, 113 times 0, 1, 141 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 528 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0, 1, 43 times 0, 1, 62 times 0, 1, 85 times 0, 1, 113 times 0, 1, 141 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
(128, 128+33, 2376922)-Net in Base 4 — Upper bound on s
There is no (128, 161, 2376923)-net in base 4, because
- 1 times m-reduction [i] would yield (128, 160, 2376923)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 135992 913974 574394 923383 695205 066216 513566 117483 136152 462754 351113 087961 650351 212040 481158 209635 > 4160 [i]