Best Known (138−18, 138, s)-Nets in Base 3
(138−18, 138, 19686)-Net over F3 — Constructive and digital
Digital (120, 138, 19686)-net over F3, using
- 1 times m-reduction [i] based on digital (120, 139, 19686)-net over F3, using
- net defined by OOA [i] based on linear OOA(3139, 19686, F3, 19, 19) (dual of [(19686, 19), 373895, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3139, 177175, F3, 19) (dual of [177175, 177036, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3139, 177176, F3, 19) (dual of [177176, 177037, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(3111, 177148, F3, 15) (dual of [177148, 177037, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3139, 177176, F3, 19) (dual of [177176, 177037, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3139, 177175, F3, 19) (dual of [177175, 177036, 20]-code), using
- net defined by OOA [i] based on linear OOA(3139, 19686, F3, 19, 19) (dual of [(19686, 19), 373895, 20]-NRT-code), using
(138−18, 138, 68006)-Net over F3 — Digital
Digital (120, 138, 68006)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3138, 68006, F3, 2, 18) (dual of [(68006, 2), 135874, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 88587, F3, 2, 18) (dual of [(88587, 2), 177036, 19]-NRT-code), using
- strength reduction [i] based on linear OOA(3138, 88587, F3, 2, 19) (dual of [(88587, 2), 177036, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3138, 177174, F3, 19) (dual of [177174, 177036, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3137, 177173, F3, 19) (dual of [177173, 177036, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3137, 177173, F3, 19) (dual of [177173, 177036, 20]-code), using
- OOA 2-folding [i] based on linear OA(3138, 177174, F3, 19) (dual of [177174, 177036, 20]-code), using
- strength reduction [i] based on linear OOA(3138, 88587, F3, 2, 19) (dual of [(88587, 2), 177036, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 88587, F3, 2, 18) (dual of [(88587, 2), 177036, 19]-NRT-code), using
(138−18, 138, large)-Net in Base 3 — Upper bound on s
There is no (120, 138, large)-net in base 3, because
- 16 times m-reduction [i] would yield (120, 122, large)-net in base 3, but