Best Known (122, 122+16, s)-Nets in Base 3
(122, 122+16, 199294)-Net over F3 — Constructive and digital
Digital (122, 138, 199294)-net over F3, using
- 33 times duplication [i] based on digital (119, 135, 199294)-net over F3, using
- net defined by OOA [i] based on linear OOA(3135, 199294, F3, 16, 16) (dual of [(199294, 16), 3188569, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3135, 1594352, F3, 16) (dual of [1594352, 1594217, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 1594353, F3, 16) (dual of [1594353, 1594218, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3135, 1594353, F3, 16) (dual of [1594353, 1594218, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3135, 1594352, F3, 16) (dual of [1594352, 1594217, 17]-code), using
- net defined by OOA [i] based on linear OOA(3135, 199294, F3, 16, 16) (dual of [(199294, 16), 3188569, 17]-NRT-code), using
(122, 122+16, 531452)-Net over F3 — Digital
Digital (122, 138, 531452)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3138, 531452, F3, 3, 16) (dual of [(531452, 3), 1594218, 17]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3135, 531451, F3, 3, 16) (dual of [(531451, 3), 1594218, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3135, 1594353, F3, 16) (dual of [1594353, 1594218, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OOA 3-folding [i] based on linear OA(3135, 1594353, F3, 16) (dual of [1594353, 1594218, 17]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3135, 531451, F3, 3, 16) (dual of [(531451, 3), 1594218, 17]-NRT-code), using
(122, 122+16, large)-Net in Base 3 — Upper bound on s
There is no (122, 138, large)-net in base 3, because
- 14 times m-reduction [i] would yield (122, 124, large)-net in base 3, but