Best Known (128−33, 128, s)-Nets in Base 4
(128−33, 128, 531)-Net over F4 — Constructive and digital
Digital (95, 128, 531)-net over F4, using
- 4 times m-reduction [i] based on digital (95, 132, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 177)-net over F64, using
(128−33, 128, 576)-Net in Base 4 — Constructive
(95, 128, 576)-net in base 4, using
- 42 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
(128−33, 128, 1117)-Net over F4 — Digital
Digital (95, 128, 1117)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4128, 1117, F4, 33) (dual of [1117, 989, 34]-code), using
- 85 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 13 times 0, 1, 22 times 0, 1, 33 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]
- 85 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 13 times 0, 1, 22 times 0, 1, 33 times 0) [i] based on linear OA(4121, 1025, F4, 33) (dual of [1025, 904, 34]-code), using
(128−33, 128, 136215)-Net in Base 4 — Upper bound on s
There is no (95, 128, 136216)-net in base 4, because
- 1 times m-reduction [i] would yield (95, 127, 136216)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 28948 432669 195151 098515 411220 927563 279842 686945 168070 861668 755222 504038 559013 > 4127 [i]