Best Known (190, 225, s)-Nets in Base 4
(190, 225, 3858)-Net over F4 — Constructive and digital
Digital (190, 225, 3858)-net over F4, using
- 45 times duplication [i] based on digital (185, 220, 3858)-net over F4, using
- net defined by OOA [i] based on linear OOA(4220, 3858, F4, 35, 35) (dual of [(3858, 35), 134810, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4220, 65587, F4, 35) (dual of [65587, 65367, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4219, 65586, F4, 35) (dual of [65586, 65367, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(410, 50, F4, 5) (dual of [50, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4219, 65586, F4, 35) (dual of [65586, 65367, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4220, 65587, F4, 35) (dual of [65587, 65367, 36]-code), using
- net defined by OOA [i] based on linear OOA(4220, 3858, F4, 35, 35) (dual of [(3858, 35), 134810, 36]-NRT-code), using
(190, 225, 53546)-Net over F4 — Digital
Digital (190, 225, 53546)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4225, 53546, F4, 35) (dual of [53546, 53321, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4225, 65554, F4, 35) (dual of [65554, 65329, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([1,17]) [i] based on
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(4208, 65537, F4, 18) (dual of [65537, 65329, 19]-code), using the narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [1,17], and minimum distance d ≥ |{−17,−15,−13,…,17}|+1 = 19 (BCH-bound) [i]
- linear OA(416, 17, F4, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,4)), using
- dual of repetition code with length 17 [i]
- construction X applied to C([0,17]) ⊂ C([1,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4225, 65554, F4, 35) (dual of [65554, 65329, 36]-code), using
(190, 225, large)-Net in Base 4 — Upper bound on s
There is no (190, 225, large)-net in base 4, because
- 33 times m-reduction [i] would yield (190, 192, large)-net in base 4, but