Best Known (119, 137, s)-Nets in Base 3
(119, 137, 19686)-Net over F3 — Constructive and digital
Digital (119, 137, 19686)-net over F3, using
- net defined by OOA [i] based on linear OOA(3137, 19686, F3, 18, 18) (dual of [(19686, 18), 354211, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3137, 177174, F3, 18) (dual of [177174, 177037, 19]-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(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 C([0,9]) ⊂ C([0,7]) [i] based on
- OA 9-folding and stacking [i] based on linear OA(3137, 177174, F3, 18) (dual of [177174, 177037, 19]-code), using
(119, 137, 63202)-Net over F3 — Digital
Digital (119, 137, 63202)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3137, 63202, F3, 2, 18) (dual of [(63202, 2), 126267, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3137, 88587, F3, 2, 18) (dual of [(88587, 2), 177037, 19]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3135, 88586, F3, 2, 18) (dual of [(88586, 2), 177037, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3135, 177172, F3, 18) (dual of [177172, 177037, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3134, 177171, F3, 18) (dual of [177171, 177037, 19]-code), using
- construction X4 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(323, 24, F3, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,3)), using
- dual of repetition code with length 24 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3134, 177171, F3, 18) (dual of [177171, 177037, 19]-code), using
- OOA 2-folding [i] based on linear OA(3135, 177172, F3, 18) (dual of [177172, 177037, 19]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3135, 88586, F3, 2, 18) (dual of [(88586, 2), 177037, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3137, 88587, F3, 2, 18) (dual of [(88587, 2), 177037, 19]-NRT-code), using
(119, 137, large)-Net in Base 3 — Upper bound on s
There is no (119, 137, large)-net in base 3, because
- 16 times m-reduction [i] would yield (119, 121, large)-net in base 3, but