Best Known (157−17, 157, s)-Nets in Base 3
(157−17, 157, 597873)-Net over F3 — Constructive and digital
Digital (140, 157, 597873)-net over F3, using
- net defined by OOA [i] based on linear OOA(3157, 597873, F3, 17, 17) (dual of [(597873, 17), 10163684, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3157, 4782985, F3, 17) (dual of [4782985, 4782828, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3155, 4782983, F3, 17) (dual of [4782983, 4782828, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3155, 4782969, F3, 17) (dual of [4782969, 4782814, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(30, 14, F3, 0) (dual of [14, 14, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3155, 4782983, F3, 17) (dual of [4782983, 4782828, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3157, 4782985, F3, 17) (dual of [4782985, 4782828, 18]-code), using
(157−17, 157, 1271815)-Net over F3 — Digital
Digital (140, 157, 1271815)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3157, 1271815, F3, 3, 17) (dual of [(1271815, 3), 3815288, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3157, 1594328, F3, 3, 17) (dual of [(1594328, 3), 4782827, 18]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3156, 1594328, F3, 3, 17) (dual of [(1594328, 3), 4782828, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3156, 4782984, F3, 17) (dual of [4782984, 4782828, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3155, 4782983, F3, 17) (dual of [4782983, 4782828, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3155, 4782969, F3, 17) (dual of [4782969, 4782814, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(30, 14, F3, 0) (dual of [14, 14, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3155, 4782983, F3, 17) (dual of [4782983, 4782828, 18]-code), using
- OOA 3-folding [i] based on linear OA(3156, 4782984, F3, 17) (dual of [4782984, 4782828, 18]-code), using
- 31 times duplication [i] based on linear OOA(3156, 1594328, F3, 3, 17) (dual of [(1594328, 3), 4782828, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3157, 1594328, F3, 3, 17) (dual of [(1594328, 3), 4782827, 18]-NRT-code), using
(157−17, 157, large)-Net in Base 3 — Upper bound on s
There is no (140, 157, large)-net in base 3, because
- 15 times m-reduction [i] would yield (140, 142, large)-net in base 3, but