Best Known (101, 101+10, s)-Nets in Base 3
(101, 101+10, 1678090)-Net over F3 — Constructive and digital
Digital (101, 111, 1678090)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (15, 20, 370)-net over F3, using
- net defined by OOA [i] based on linear OOA(320, 370, F3, 5, 5) (dual of [(370, 5), 1830, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(320, 741, F3, 5) (dual of [741, 721, 6]-code), using
- construction XX applied to C1 = C([361,364]), C2 = C([363,365]), C3 = C1 + C2 = C([363,364]), and C∩ = C1 ∩ C2 = C([361,365]) [i] based on
- linear OA(313, 728, F3, 4) (dual of [728, 715, 5]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {361,362,363,364}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 728, F3, 3) (dual of [728, 715, 4]-code or 728-cap in PG(12,3)), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {363,364,365}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(319, 728, F3, 5) (dual of [728, 709, 6]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {361,362,363,364,365}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(37, 728, F3, 2) (dual of [728, 721, 3]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {363,364}, and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([361,364]), C2 = C([363,365]), C3 = C1 + C2 = C([363,364]), and C∩ = C1 ∩ C2 = C([361,365]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(320, 741, F3, 5) (dual of [741, 721, 6]-code), using
- net defined by OOA [i] based on linear OOA(320, 370, F3, 5, 5) (dual of [(370, 5), 1830, 6]-NRT-code), using
- digital (81, 91, 1677720)-net over F3, using
- net defined by OOA [i] based on linear OOA(391, 1677720, F3, 10, 10) (dual of [(1677720, 10), 16777109, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(391, 8388600, F3, 10) (dual of [8388600, 8388509, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(391, 8388600, F3, 10) (dual of [8388600, 8388509, 11]-code), using
- net defined by OOA [i] based on linear OOA(391, 1677720, F3, 10, 10) (dual of [(1677720, 10), 16777109, 11]-NRT-code), using
- digital (15, 20, 370)-net over F3, using
(101, 101+10, 6840327)-Net over F3 — Digital
Digital (101, 111, 6840327)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3111, 6840327, F3, 10) (dual of [6840327, 6840216, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3111, large, F3, 10) (dual of [large, large−111, 11]-code), using
- 20 times code embedding in larger space [i] based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 20 times code embedding in larger space [i] based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3111, large, F3, 10) (dual of [large, large−111, 11]-code), using
(101, 101+10, large)-Net in Base 3 — Upper bound on s
There is no (101, 111, large)-net in base 3, because
- 8 times m-reduction [i] would yield (101, 103, large)-net in base 3, but