Best Known (87, 87+20, s)-Nets in Base 3
(87, 87+20, 657)-Net over F3 — Constructive and digital
Digital (87, 107, 657)-net over F3, using
- 31 times duplication [i] based on digital (86, 106, 657)-net over F3, using
- net defined by OOA [i] based on linear OOA(3106, 657, F3, 20, 20) (dual of [(657, 20), 13034, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3106, 6570, F3, 20) (dual of [6570, 6464, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3105, 6569, F3, 20) (dual of [6569, 6464, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3105, 6569, F3, 20) (dual of [6569, 6464, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3106, 6570, F3, 20) (dual of [6570, 6464, 21]-code), using
- net defined by OOA [i] based on linear OOA(3106, 657, F3, 20, 20) (dual of [(657, 20), 13034, 21]-NRT-code), using
(87, 87+20, 3160)-Net over F3 — Digital
Digital (87, 107, 3160)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3107, 3160, F3, 2, 20) (dual of [(3160, 2), 6213, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3107, 3286, F3, 2, 20) (dual of [(3286, 2), 6465, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3107, 6572, F3, 20) (dual of [6572, 6465, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3107, 6572, F3, 20) (dual of [6572, 6465, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3107, 3286, F3, 2, 20) (dual of [(3286, 2), 6465, 21]-NRT-code), using
(87, 87+20, 288489)-Net in Base 3 — Upper bound on s
There is no (87, 107, 288490)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1127 165194 788330 544897 837056 133029 953483 501032 834197 > 3107 [i]