Best Known (42−12, 42, s)-Nets in Base 4
(42−12, 42, 240)-Net over F4 — Constructive and digital
Digital (30, 42, 240)-net over F4, using
- t-expansion [i] based on digital (29, 42, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 14, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 14, 80)-net over F64, using
(42−12, 42, 337)-Net over F4 — Digital
Digital (30, 42, 337)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(442, 337, F4, 12) (dual of [337, 295, 13]-code), using
- 76 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 21 times 0, 1, 30 times 0) [i] based on linear OA(436, 255, F4, 12) (dual of [255, 219, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 76 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 21 times 0, 1, 30 times 0) [i] based on linear OA(436, 255, F4, 12) (dual of [255, 219, 13]-code), using
(42−12, 42, 387)-Net in Base 4 — Constructive
(30, 42, 387)-net in base 4, using
- trace code for nets [i] based on (2, 14, 129)-net in base 64, using
- base change [i] based on digital (0, 12, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 12, 129)-net over F128, using
(42−12, 42, 16345)-Net in Base 4 — Upper bound on s
There is no (30, 42, 16346)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 19 345554 417109 055878 810684 > 442 [i]