Best Known (160−20, 160, s)-Nets in Base 4
(160−20, 160, 104862)-Net over F4 — Constructive and digital
Digital (140, 160, 104862)-net over F4, using
- 1 times m-reduction [i] based on digital (140, 161, 104862)-net over F4, using
- net defined by OOA [i] based on linear OOA(4161, 104862, F4, 21, 21) (dual of [(104862, 21), 2201941, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4161, 1048621, F4, 21) (dual of [1048621, 1048460, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4161, 1048626, F4, 21) (dual of [1048626, 1048465, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4161, 1048626, F4, 21) (dual of [1048626, 1048465, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4161, 1048621, F4, 21) (dual of [1048621, 1048460, 22]-code), using
- net defined by OOA [i] based on linear OOA(4161, 104862, F4, 21, 21) (dual of [(104862, 21), 2201941, 22]-NRT-code), using
(160−20, 160, 524313)-Net over F4 — Digital
Digital (140, 160, 524313)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4160, 524313, F4, 2, 20) (dual of [(524313, 2), 1048466, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4160, 1048626, F4, 20) (dual of [1048626, 1048466, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4160, 1048627, F4, 20) (dual of [1048627, 1048467, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4160, 1048627, F4, 20) (dual of [1048627, 1048467, 21]-code), using
- OOA 2-folding [i] based on linear OA(4160, 1048626, F4, 20) (dual of [1048626, 1048466, 21]-code), using
(160−20, 160, large)-Net in Base 4 — Upper bound on s
There is no (140, 160, large)-net in base 4, because
- 18 times m-reduction [i] would yield (140, 142, large)-net in base 4, but