Best Known (151−16, 151, s)-Nets in Base 4
(151−16, 151, 1048575)-Net over F4 — Constructive and digital
Digital (135, 151, 1048575)-net over F4, using
- 46 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
(151−16, 151, 5688247)-Net over F4 — Digital
Digital (135, 151, 5688247)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4151, 5688247, F4, 16) (dual of [5688247, 5688096, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4151, large, F4, 16) (dual of [large, large−151, 17]-code), using
- 7 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]
- 7 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(4151, large, F4, 16) (dual of [large, large−151, 17]-code), using
(151−16, 151, large)-Net in Base 4 — Upper bound on s
There is no (135, 151, large)-net in base 4, because
- 14 times m-reduction [i] would yield (135, 137, large)-net in base 4, but