Best Known (211−31, 211, s)-Nets in Base 3
(211−31, 211, 3939)-Net over F3 — Constructive and digital
Digital (180, 211, 3939)-net over F3, using
- 32 times duplication [i] based on digital (178, 209, 3939)-net over F3, using
- net defined by OOA [i] based on linear OOA(3209, 3939, F3, 31, 31) (dual of [(3939, 31), 121900, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3209, 59086, F3, 31) (dual of [59086, 58877, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3209, 59087, F3, 31) (dual of [59087, 58878, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3209, 59087, F3, 31) (dual of [59087, 58878, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3209, 59086, F3, 31) (dual of [59086, 58877, 32]-code), using
- net defined by OOA [i] based on linear OOA(3209, 3939, F3, 31, 31) (dual of [(3939, 31), 121900, 32]-NRT-code), using
(211−31, 211, 20548)-Net over F3 — Digital
Digital (180, 211, 20548)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3211, 20548, F3, 2, 31) (dual of [(20548, 2), 40885, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3211, 29544, F3, 2, 31) (dual of [(29544, 2), 58877, 32]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3210, 29544, F3, 2, 31) (dual of [(29544, 2), 58878, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3210, 59088, F3, 31) (dual of [59088, 58878, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3209, 59087, F3, 31) (dual of [59087, 58878, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3209, 59087, F3, 31) (dual of [59087, 58878, 32]-code), using
- OOA 2-folding [i] based on linear OA(3210, 59088, F3, 31) (dual of [59088, 58878, 32]-code), using
- 31 times duplication [i] based on linear OOA(3210, 29544, F3, 2, 31) (dual of [(29544, 2), 58878, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3211, 29544, F3, 2, 31) (dual of [(29544, 2), 58877, 32]-NRT-code), using
(211−31, 211, large)-Net in Base 3 — Upper bound on s
There is no (180, 211, large)-net in base 3, because
- 29 times m-reduction [i] would yield (180, 182, large)-net in base 3, but