Best Known (134, 134+16, s)-Nets in Base 4
(134, 134+16, 1048575)-Net over F4 — Constructive and digital
Digital (134, 150, 1048575)-net over F4, using
- 45 times duplication [i] based on digital (129, 145, 1048575)-net over F4, using
- t-expansion [i] based on digital (128, 145, 1048575)-net over F4, using
- net defined by OOA [i] based on linear OOA(4145, 1048575, F4, 17, 17) (dual of [(1048575, 17), 17825630, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4145, 8388601, F4, 17) (dual of [8388601, 8388456, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4145, 8388601, F4, 17) (dual of [8388601, 8388456, 18]-code), using
- net defined by OOA [i] based on linear OOA(4145, 1048575, F4, 17, 17) (dual of [(1048575, 17), 17825630, 18]-NRT-code), using
- t-expansion [i] based on digital (128, 145, 1048575)-net over F4, using
(134, 134+16, 5151979)-Net over F4 — Digital
Digital (134, 150, 5151979)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4150, 5151979, F4, 16) (dual of [5151979, 5151829, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4150, large, F4, 16) (dual of [large, large−150, 17]-code), using
- 6 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 6 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4150, large, F4, 16) (dual of [large, large−150, 17]-code), using
(134, 134+16, large)-Net in Base 4 — Upper bound on s
There is no (134, 150, large)-net in base 4, because
- 14 times m-reduction [i] would yield (134, 136, large)-net in base 4, but