Best Known (136, 136+12, s)-Nets in Base 3
(136, 136+12, 1594329)-Net over F3 — Constructive and digital
Digital (136, 148, 1594329)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (130, 142, 1594325)-net over F3, using
- net defined by OOA [i] based on linear OOA(3142, 1594325, F3, 14, 12) (dual of [(1594325, 14), 22320408, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3142, 4782976, F3, 2, 12) (dual of [(4782976, 2), 9565810, 13]-NRT-code), using
- trace code [i] based on linear OOA(971, 2391488, F9, 2, 12) (dual of [(2391488, 2), 4782905, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(971, 4782976, F9, 12) (dual of [4782976, 4782905, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(971, 4782976, F9, 12) (dual of [4782976, 4782905, 13]-code), using
- trace code [i] based on linear OOA(971, 2391488, F9, 2, 12) (dual of [(2391488, 2), 4782905, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3142, 4782976, F3, 2, 12) (dual of [(4782976, 2), 9565810, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3142, 1594325, F3, 14, 12) (dual of [(1594325, 14), 22320408, 13]-NRT-code), using
- digital (0, 6, 4)-net over F3, using
(136, 136+12, large)-Net over F3 — Digital
Digital (136, 148, large)-net over F3, using
- 39 times duplication [i] based on digital (127, 139, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3139, large, F3, 12) (dual of [large, large−139, 13]-code), using
- 19 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 19 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3139, large, F3, 12) (dual of [large, large−139, 13]-code), using
(136, 136+12, large)-Net in Base 3 — Upper bound on s
There is no (136, 148, large)-net in base 3, because
- 10 times m-reduction [i] would yield (136, 138, large)-net in base 3, but