Best Known (127, 127+11, s)-Nets in Base 3
(127, 127+11, 1913228)-Net over F3 — Constructive and digital
Digital (127, 138, 1913228)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 10, 38)-net over F3, using
- digital (117, 128, 1913190)-net over F3, using
- trace code for nets [i] based on digital (53, 64, 956595)-net over F9, using
- net defined by OOA [i] based on linear OOA(964, 956595, F9, 11, 11) (dual of [(956595, 11), 10522481, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(964, 4782976, F9, 11) (dual of [4782976, 4782912, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(964, 4782976, F9, 11) (dual of [4782976, 4782912, 12]-code), using
- net defined by OOA [i] based on linear OOA(964, 956595, F9, 11, 11) (dual of [(956595, 11), 10522481, 12]-NRT-code), using
- trace code for nets [i] based on digital (53, 64, 956595)-net over F9, using
(127, 127+11, large)-Net over F3 — Digital
Digital (127, 138, large)-net over F3, using
- 1 times m-reduction [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
(127, 127+11, large)-Net in Base 3 — Upper bound on s
There is no (127, 138, large)-net in base 3, because
- 9 times m-reduction [i] would yield (127, 129, large)-net in base 3, but