Best Known (172−38, 172, s)-Nets in Base 4
(172−38, 172, 1048)-Net over F4 — Constructive and digital
Digital (134, 172, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 43, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
(172−38, 172, 3418)-Net over F4 — Digital
Digital (134, 172, 3418)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4172, 3418, F4, 38) (dual of [3418, 3246, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 4111, F4, 38) (dual of [4111, 3939, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(34) [i] based on
- linear OA(4169, 4096, F4, 38) (dual of [4096, 3927, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(43, 15, F4, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(37) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4172, 4111, F4, 38) (dual of [4111, 3939, 39]-code), using
(172−38, 172, 745268)-Net in Base 4 — Upper bound on s
There is no (134, 172, 745269)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 35 836621 933376 531577 483813 324234 357454 280577 214605 802949 908825 619946 507680 397280 658729 850608 670531 625948 > 4172 [i]