Best Known (168−13, 168, s)-Nets in Base 3
(168−13, 168, 1594358)-Net over F3 — Constructive and digital
Digital (155, 168, 1594358)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 33)-net over F3, using
- digital (143, 156, 1594325)-net over F3, using
- net defined by OOA [i] based on linear OOA(3156, 1594325, F3, 14, 13) (dual of [(1594325, 14), 22320394, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3156, 4782976, F3, 2, 13) (dual of [(4782976, 2), 9565796, 14]-NRT-code), using
- trace code [i] based on linear OOA(978, 2391488, F9, 2, 13) (dual of [(2391488, 2), 4782898, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(978, 4782976, F9, 13) (dual of [4782976, 4782898, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(978, 4782976, F9, 13) (dual of [4782976, 4782898, 14]-code), using
- trace code [i] based on linear OOA(978, 2391488, F9, 2, 13) (dual of [(2391488, 2), 4782898, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3156, 4782976, F3, 2, 13) (dual of [(4782976, 2), 9565796, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3156, 1594325, F3, 14, 13) (dual of [(1594325, 14), 22320394, 14]-NRT-code), using
(168−13, 168, large)-Net over F3 — Digital
Digital (155, 168, large)-net over F3, using
- 33 times duplication [i] based on digital (152, 165, large)-net over F3, using
- t-expansion [i] based on digital (151, 165, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- t-expansion [i] based on digital (151, 165, large)-net over F3, using
(168−13, 168, large)-Net in Base 3 — Upper bound on s
There is no (155, 168, large)-net in base 3, because
- 11 times m-reduction [i] would yield (155, 157, large)-net in base 3, but