Best Known (187−25, 187, s)-Nets in Base 3
(187−25, 187, 14765)-Net over F3 — Constructive and digital
Digital (162, 187, 14765)-net over F3, using
- 32 times duplication [i] based on digital (160, 185, 14765)-net over F3, using
- net defined by OOA [i] based on linear OOA(3185, 14765, F3, 25, 25) (dual of [(14765, 25), 368940, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3185, 177181, F3, 25) (dual of [177181, 176996, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3185, 177188, F3, 25) (dual of [177188, 177003, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3185, 177188, F3, 25) (dual of [177188, 177003, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3185, 177181, F3, 25) (dual of [177181, 176996, 26]-code), using
- net defined by OOA [i] based on linear OOA(3185, 14765, F3, 25, 25) (dual of [(14765, 25), 368940, 26]-NRT-code), using
(187−25, 187, 59063)-Net over F3 — Digital
Digital (162, 187, 59063)-net over F3, using
- 31 times duplication [i] based on digital (161, 186, 59063)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3186, 59063, F3, 3, 25) (dual of [(59063, 3), 177003, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3186, 177189, F3, 25) (dual of [177189, 177003, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3185, 177188, F3, 25) (dual of [177188, 177003, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3185, 177188, F3, 25) (dual of [177188, 177003, 26]-code), using
- OOA 3-folding [i] based on linear OA(3186, 177189, F3, 25) (dual of [177189, 177003, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3186, 59063, F3, 3, 25) (dual of [(59063, 3), 177003, 26]-NRT-code), using
(187−25, 187, large)-Net in Base 3 — Upper bound on s
There is no (162, 187, large)-net in base 3, because
- 23 times m-reduction [i] would yield (162, 164, large)-net in base 3, but