Best Known (53−12, 53, s)-Nets in Base 4
(53−12, 53, 1032)-Net over F4 — Constructive and digital
Digital (41, 53, 1032)-net over F4, using
- 41 times duplication [i] based on digital (40, 52, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 13, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 13, 258)-net over F256, using
(53−12, 53, 1337)-Net over F4 — Digital
Digital (41, 53, 1337)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 1337, F4, 12) (dual of [1337, 1284, 13]-code), using
- 306 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 14 times 0, 1, 27 times 0, 1, 50 times 0, 1, 82 times 0, 1, 119 times 0) [i] based on linear OA(445, 1023, F4, 12) (dual of [1023, 978, 13]-code), using
- 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]
- 306 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 14 times 0, 1, 27 times 0, 1, 50 times 0, 1, 82 times 0, 1, 119 times 0) [i] based on linear OA(445, 1023, F4, 12) (dual of [1023, 978, 13]-code), using
(53−12, 53, 207628)-Net in Base 4 — Upper bound on s
There is no (41, 53, 207629)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 81 129645 808663 528942 142693 614732 > 453 [i]