Best Known (131, 131+16, s)-Nets in Base 3
(131, 131+16, 597875)-Net over F3 — Constructive and digital
Digital (131, 147, 597875)-net over F3, using
- 32 times duplication [i] based on digital (129, 145, 597875)-net over F3, using
- net defined by OOA [i] based on linear OOA(3145, 597875, F3, 16, 16) (dual of [(597875, 16), 9565855, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3145, 4783000, F3, 16) (dual of [4783000, 4782855, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3145, 4783001, F3, 16) (dual of [4783001, 4782856, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 32, F3, 2) (dual of [32, 28, 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(3145, 4783001, F3, 16) (dual of [4783001, 4782856, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3145, 4783000, F3, 16) (dual of [4783000, 4782855, 17]-code), using
- net defined by OOA [i] based on linear OOA(3145, 597875, F3, 16, 16) (dual of [(597875, 16), 9565855, 17]-NRT-code), using
(131, 131+16, 1405345)-Net over F3 — Digital
Digital (131, 147, 1405345)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 1405345, F3, 3, 16) (dual of [(1405345, 3), 4215888, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3147, 1594334, F3, 3, 16) (dual of [(1594334, 3), 4782855, 17]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3146, 1594334, F3, 3, 16) (dual of [(1594334, 3), 4782856, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3146, 4783002, F3, 16) (dual of [4783002, 4782856, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3145, 4783001, F3, 16) (dual of [4783001, 4782856, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 32, F3, 2) (dual of [32, 28, 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
- 1 times code embedding in larger space [i] based on linear OA(3145, 4783001, F3, 16) (dual of [4783001, 4782856, 17]-code), using
- OOA 3-folding [i] based on linear OA(3146, 4783002, F3, 16) (dual of [4783002, 4782856, 17]-code), using
- 31 times duplication [i] based on linear OOA(3146, 1594334, F3, 3, 16) (dual of [(1594334, 3), 4782856, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3147, 1594334, F3, 3, 16) (dual of [(1594334, 3), 4782855, 17]-NRT-code), using
(131, 131+16, large)-Net in Base 3 — Upper bound on s
There is no (131, 147, large)-net in base 3, because
- 14 times m-reduction [i] would yield (131, 133, large)-net in base 3, but