Best Known (40, 40+13, s)-Nets in Base 4
(40, 40+13, 1028)-Net over F4 — Constructive and digital
Digital (40, 53, 1028)-net over F4, using
- 41 times duplication [i] based on digital (39, 52, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 257)-net over F256, using
(40, 40+13, 1072)-Net over F4 — Digital
Digital (40, 53, 1072)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 1072, F4, 13) (dual of [1072, 1019, 14]-code), using
- 41 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0) [i] based on linear OA(446, 1024, F4, 13) (dual of [1024, 978, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 41 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0) [i] based on linear OA(446, 1024, F4, 13) (dual of [1024, 978, 14]-code), using
(40, 40+13, 164794)-Net in Base 4 — Upper bound on s
There is no (40, 53, 164795)-net in base 4, because
- 1 times m-reduction [i] would yield (40, 52, 164795)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 20 282914 942190 778204 326909 217358 > 452 [i]