Best Known (125−11, 125, s)-Nets in Base 3
(125−11, 125, 1678087)-Net over F3 — Constructive and digital
Digital (114, 125, 1678087)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (14, 19, 367)-net over F3, using
- net defined by OOA [i] based on linear OOA(319, 367, F3, 5, 5) (dual of [(367, 5), 1816, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(319, 735, F3, 5) (dual of [735, 716, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(319, 729, F3, 5) (dual of [729, 710, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 729, F3, 4) (dual of [729, 716, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(319, 735, F3, 5) (dual of [735, 716, 6]-code), using
- net defined by OOA [i] based on linear OOA(319, 367, F3, 5, 5) (dual of [(367, 5), 1816, 6]-NRT-code), using
- digital (95, 106, 1677720)-net over F3, using
- net defined by OOA [i] based on linear OOA(3106, 1677720, F3, 11, 11) (dual of [(1677720, 11), 18454814, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3106, 8388601, F3, 11) (dual of [8388601, 8388495, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3106, 8388601, F3, 11) (dual of [8388601, 8388495, 12]-code), using
- net defined by OOA [i] based on linear OOA(3106, 1677720, F3, 11, 11) (dual of [(1677720, 11), 18454814, 12]-NRT-code), using
- digital (14, 19, 367)-net over F3, using
(125−11, 125, 7769474)-Net over F3 — Digital
Digital (114, 125, 7769474)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3125, 7769474, F3, 11) (dual of [7769474, 7769349, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, large, F3, 11) (dual of [large, large−125, 12]-code), using
- 19 times code embedding in larger space [i] based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 19 times code embedding in larger space [i] based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, large, F3, 11) (dual of [large, large−125, 12]-code), using
(125−11, 125, large)-Net in Base 3 — Upper bound on s
There is no (114, 125, large)-net in base 3, because
- 9 times m-reduction [i] would yield (114, 116, large)-net in base 3, but