Best Known (137, 154, s)-Nets in Base 3
(137, 154, 199299)-Net over F3 — Constructive and digital
Digital (137, 154, 199299)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (127, 144, 199291)-net over F3, using
- net defined by OOA [i] based on linear OOA(3144, 199291, F3, 17, 17) (dual of [(199291, 17), 3387803, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3144, 1594329, F3, 17) (dual of [1594329, 1594185, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 1594336, F3, 17) (dual of [1594336, 1594192, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(30, 13, F3, 0) (dual of [13, 13, 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
- discarding factors / shortening the dual code based on linear OA(3144, 1594336, F3, 17) (dual of [1594336, 1594192, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3144, 1594329, F3, 17) (dual of [1594329, 1594185, 18]-code), using
- net defined by OOA [i] based on linear OOA(3144, 199291, F3, 17, 17) (dual of [(199291, 17), 3387803, 18]-NRT-code), using
- digital (2, 10, 8)-net over F3, using
(137, 154, 531457)-Net over F3 — Digital
Digital (137, 154, 531457)-net over F3, using
- 31 times duplication [i] based on digital (136, 153, 531457)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 531457, F3, 3, 17) (dual of [(531457, 3), 1594218, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3150, 531456, F3, 3, 17) (dual of [(531456, 3), 1594218, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3150, 1594368, F3, 17) (dual of [1594368, 1594218, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- OOA 3-folding [i] based on linear OA(3150, 1594368, F3, 17) (dual of [1594368, 1594218, 18]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3150, 531456, F3, 3, 17) (dual of [(531456, 3), 1594218, 18]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 531457, F3, 3, 17) (dual of [(531457, 3), 1594218, 18]-NRT-code), using
(137, 154, large)-Net in Base 3 — Upper bound on s
There is no (137, 154, large)-net in base 3, because
- 15 times m-reduction [i] would yield (137, 139, large)-net in base 3, but