Best Known (127−33, 127, s)-Nets in Base 4
(127−33, 127, 531)-Net over F4 — Constructive and digital
Digital (94, 127, 531)-net over F4, using
- t-expansion [i] based on digital (93, 127, 531)-net over F4, using
- 2 times m-reduction [i] based on digital (93, 129, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 43, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 43, 177)-net over F64, using
- 2 times m-reduction [i] based on digital (93, 129, 531)-net over F4, using
(127−33, 127, 576)-Net in Base 4 — Constructive
(94, 127, 576)-net in base 4, using
- 41 times duplication [i] based on (93, 126, 576)-net in base 4, using
- trace code for nets [i] based on (9, 42, 192)-net in base 64, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- trace code for nets [i] based on (9, 42, 192)-net in base 64, using
(127−33, 127, 1082)-Net over F4 — Digital
Digital (94, 127, 1082)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4127, 1082, F4, 33) (dual of [1082, 955, 34]-code), using
- 51 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 13 times 0, 1, 22 times 0) [i] based on linear OA(4121, 1025, F4, 33) (dual of [1025, 904, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 51 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 13 times 0, 1, 22 times 0) [i] based on linear OA(4121, 1025, F4, 33) (dual of [1025, 904, 34]-code), using
(127−33, 127, 124909)-Net in Base 4 — Upper bound on s
There is no (94, 127, 124910)-net in base 4, because
- 1 times m-reduction [i] would yield (94, 126, 124910)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7237 518142 585797 707758 414383 287470 261647 961930 054771 082243 375359 066970 849878 > 4126 [i]