Best Known (131, 131+23, s)-Nets in Base 3
(131, 131+23, 5369)-Net over F3 — Constructive and digital
Digital (131, 154, 5369)-net over F3, using
- 32 times duplication [i] based on digital (129, 152, 5369)-net over F3, using
- net defined by OOA [i] based on linear OOA(3152, 5369, F3, 23, 23) (dual of [(5369, 23), 123335, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3152, 59060, F3, 23) (dual of [59060, 58908, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3151, 59059, F3, 23) (dual of [59059, 58908, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(22) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3151, 59059, F3, 23) (dual of [59059, 58908, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3152, 59060, F3, 23) (dual of [59060, 58908, 24]-code), using
- net defined by OOA [i] based on linear OOA(3152, 5369, F3, 23, 23) (dual of [(5369, 23), 123335, 24]-NRT-code), using
(131, 131+23, 19687)-Net over F3 — Digital
Digital (131, 154, 19687)-net over F3, using
- 31 times duplication [i] based on digital (130, 153, 19687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 19687, F3, 3, 23) (dual of [(19687, 3), 58908, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3153, 59061, F3, 23) (dual of [59061, 58908, 24]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3151, 59059, F3, 23) (dual of [59059, 58908, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(22) ⊂ Ce(21) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3151, 59059, F3, 23) (dual of [59059, 58908, 24]-code), using
- OOA 3-folding [i] based on linear OA(3153, 59061, F3, 23) (dual of [59061, 58908, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 19687, F3, 3, 23) (dual of [(19687, 3), 58908, 24]-NRT-code), using
(131, 131+23, large)-Net in Base 3 — Upper bound on s
There is no (131, 154, large)-net in base 3, because
- 21 times m-reduction [i] would yield (131, 133, large)-net in base 3, but