Best Known (190, 190+37, s)-Nets in Base 4
(190, 190+37, 3643)-Net over F4 — Constructive and digital
Digital (190, 227, 3643)-net over F4, using
- net defined by OOA [i] based on linear OOA(4227, 3643, F4, 37, 37) (dual of [(3643, 37), 134564, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4227, 65575, F4, 37) (dual of [65575, 65348, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4227, 65578, F4, 37) (dual of [65578, 65351, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(410, 42, F4, 5) (dual of [42, 32, 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(36) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4227, 65578, F4, 37) (dual of [65578, 65351, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4227, 65575, F4, 37) (dual of [65575, 65348, 38]-code), using
(190, 190+37, 35759)-Net over F4 — Digital
Digital (190, 227, 35759)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4227, 35759, F4, 37) (dual of [35759, 35532, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4227, 65578, F4, 37) (dual of [65578, 65351, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(410, 42, F4, 5) (dual of [42, 32, 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(36) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4227, 65578, F4, 37) (dual of [65578, 65351, 38]-code), using
(190, 190+37, large)-Net in Base 4 — Upper bound on s
There is no (190, 227, large)-net in base 4, because
- 35 times m-reduction [i] would yield (190, 192, large)-net in base 4, but