Best Known (231−37, 231, s)-Nets in Base 4
(231−37, 231, 3643)-Net over F4 — Constructive and digital
Digital (194, 231, 3643)-net over F4, using
- 44 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(4227, 3643, F4, 37, 37) (dual of [(3643, 37), 134564, 38]-NRT-code), using
(231−37, 231, 41903)-Net over F4 — Digital
Digital (194, 231, 41903)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4231, 41903, F4, 37) (dual of [41903, 41672, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 65590, F4, 37) (dual of [65590, 65359, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4230, 65589, F4, 37) (dual of [65589, 65359, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(29) [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(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(36) ⊂ Ce(29) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4230, 65589, F4, 37) (dual of [65589, 65359, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 65590, F4, 37) (dual of [65590, 65359, 38]-code), using
(231−37, 231, large)-Net in Base 4 — Upper bound on s
There is no (194, 231, large)-net in base 4, because
- 35 times m-reduction [i] would yield (194, 196, large)-net in base 4, but