Best Known (223−32, 223, s)-Nets in Base 3
(223−32, 223, 3693)-Net over F3 — Constructive and digital
Digital (191, 223, 3693)-net over F3, using
- 33 times duplication [i] based on digital (188, 220, 3693)-net over F3, using
- net defined by OOA [i] based on linear OOA(3220, 3693, F3, 32, 32) (dual of [(3693, 32), 117956, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3220, 59088, F3, 32) (dual of [59088, 58868, 33]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3217, 59085, F3, 32) (dual of [59085, 58868, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3217, 59085, F3, 32) (dual of [59085, 58868, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3220, 59088, F3, 32) (dual of [59088, 58868, 33]-code), using
- net defined by OOA [i] based on linear OOA(3220, 3693, F3, 32, 32) (dual of [(3693, 32), 117956, 33]-NRT-code), using
(223−32, 223, 25212)-Net over F3 — Digital
Digital (191, 223, 25212)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3223, 25212, F3, 2, 32) (dual of [(25212, 2), 50201, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3223, 29550, F3, 2, 32) (dual of [(29550, 2), 58877, 33]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3222, 29550, F3, 2, 32) (dual of [(29550, 2), 58878, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3222, 59100, F3, 32) (dual of [59100, 58878, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3171, 59049, F3, 26) (dual of [59049, 58878, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(3222, 59100, F3, 32) (dual of [59100, 58878, 33]-code), using
- 31 times duplication [i] based on linear OOA(3222, 29550, F3, 2, 32) (dual of [(29550, 2), 58878, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3223, 29550, F3, 2, 32) (dual of [(29550, 2), 58877, 33]-NRT-code), using
(223−32, 223, large)-Net in Base 3 — Upper bound on s
There is no (191, 223, large)-net in base 3, because
- 30 times m-reduction [i] would yield (191, 193, large)-net in base 3, but