Best Known (143−16, 143, s)-Nets in Base 3
(143−16, 143, 597873)-Net over F3 — Constructive and digital
Digital (127, 143, 597873)-net over F3, using
- 31 times duplication [i] based on digital (126, 142, 597873)-net over F3, using
- net defined by OOA [i] based on linear OOA(3142, 597873, F3, 16, 16) (dual of [(597873, 16), 9565826, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3142, 4782984, F3, 16) (dual of [4782984, 4782842, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(3127, 4782969, F3, 14) (dual of [4782969, 4782842, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- OA 8-folding and stacking [i] based on linear OA(3142, 4782984, F3, 16) (dual of [4782984, 4782842, 17]-code), using
- net defined by OOA [i] based on linear OOA(3142, 597873, F3, 16, 16) (dual of [(597873, 16), 9565826, 17]-NRT-code), using
(143−16, 143, 1195746)-Net over F3 — Digital
Digital (127, 143, 1195746)-net over F3, using
- 31 times duplication [i] based on digital (126, 142, 1195746)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3142, 1195746, F3, 4, 16) (dual of [(1195746, 4), 4782842, 17]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3142, 4782984, F3, 16) (dual of [4782984, 4782842, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(3127, 4782969, F3, 14) (dual of [4782969, 4782842, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- OOA 4-folding [i] based on linear OA(3142, 4782984, F3, 16) (dual of [4782984, 4782842, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3142, 1195746, F3, 4, 16) (dual of [(1195746, 4), 4782842, 17]-NRT-code), using
(143−16, 143, large)-Net in Base 3 — Upper bound on s
There is no (127, 143, large)-net in base 3, because
- 14 times m-reduction [i] would yield (127, 129, large)-net in base 3, but