Best Known (61, 61+14, s)-Nets in Base 3
(61, 61+14, 938)-Net over F3 — Constructive and digital
Digital (61, 75, 938)-net over F3, using
- 32 times duplication [i] based on digital (59, 73, 938)-net over F3, using
- net defined by OOA [i] based on linear OOA(373, 938, F3, 14, 14) (dual of [(938, 14), 13059, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(373, 6566, F3, 14) (dual of [6566, 6493, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(373, 6566, F3, 14) (dual of [6566, 6493, 15]-code), using
- net defined by OOA [i] based on linear OOA(373, 938, F3, 14, 14) (dual of [(938, 14), 13059, 15]-NRT-code), using
(61, 61+14, 3286)-Net over F3 — Digital
Digital (61, 75, 3286)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(375, 3286, F3, 2, 14) (dual of [(3286, 2), 6497, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(375, 6572, F3, 14) (dual of [6572, 6497, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(375, 6572, F3, 14) (dual of [6572, 6497, 15]-code), using
(61, 61+14, 218720)-Net in Base 3 — Upper bound on s
There is no (61, 75, 218721)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 608279 465741 390927 913797 448748 261899 > 375 [i]