Best Known (138, 138+23, s)-Nets in Base 3
(138, 138+23, 5371)-Net over F3 — Constructive and digital
Digital (138, 161, 5371)-net over F3, using
- 34 times duplication [i] based on digital (134, 157, 5371)-net over F3, using
- net defined by OOA [i] based on linear OOA(3157, 5371, F3, 23, 23) (dual of [(5371, 23), 123376, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3157, 59082, F3, 23) (dual of [59082, 58925, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 59085, F3, 23) (dual of [59085, 58928, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3157, 59085, F3, 23) (dual of [59085, 58928, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3157, 59082, F3, 23) (dual of [59082, 58925, 24]-code), using
- net defined by OOA [i] based on linear OOA(3157, 5371, F3, 23, 23) (dual of [(5371, 23), 123376, 24]-NRT-code), using
(138, 138+23, 25767)-Net over F3 — Digital
Digital (138, 161, 25767)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3161, 25767, F3, 2, 23) (dual of [(25767, 2), 51373, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3161, 29544, F3, 2, 23) (dual of [(29544, 2), 58927, 24]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3160, 29544, F3, 2, 23) (dual of [(29544, 2), 58928, 24]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3158, 29543, F3, 2, 23) (dual of [(29543, 2), 58928, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3158, 59086, F3, 23) (dual of [59086, 58928, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3157, 59085, F3, 23) (dual of [59085, 58928, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(22) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3157, 59085, F3, 23) (dual of [59085, 58928, 24]-code), using
- OOA 2-folding [i] based on linear OA(3158, 59086, F3, 23) (dual of [59086, 58928, 24]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3158, 29543, F3, 2, 23) (dual of [(29543, 2), 58928, 24]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3160, 29544, F3, 2, 23) (dual of [(29544, 2), 58928, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3161, 29544, F3, 2, 23) (dual of [(29544, 2), 58927, 24]-NRT-code), using
(138, 138+23, large)-Net in Base 3 — Upper bound on s
There is no (138, 161, large)-net in base 3, because
- 21 times m-reduction [i] would yield (138, 140, large)-net in base 3, but