Best Known (16, 16+10, s)-Nets in Base 27
(16, 16+10, 292)-Net over F27 — Constructive and digital
Digital (16, 26, 292)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- digital (9, 19, 146)-net over F27, using
- net defined by OOA [i] based on linear OOA(2719, 146, F27, 10, 10) (dual of [(146, 10), 1441, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2719, 730, F27, 10) (dual of [730, 711, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2719, 731, F27, 10) (dual of [731, 712, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(2719, 729, F27, 10) (dual of [729, 710, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2717, 729, F27, 9) (dual of [729, 712, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(2719, 731, F27, 10) (dual of [731, 712, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2719, 730, F27, 10) (dual of [730, 711, 11]-code), using
- net defined by OOA [i] based on linear OOA(2719, 146, F27, 10, 10) (dual of [(146, 10), 1441, 11]-NRT-code), using
- digital (2, 7, 351)-net over F27, using
(16, 16+10, 1312)-Net in Base 27 — Constructive
(16, 26, 1312)-net in base 27, using
- net defined by OOA [i] based on OOA(2726, 1312, S27, 10, 10), using
- OA 5-folding and stacking [i] based on OA(2726, 6560, S27, 10), using
- discarding factors based on OA(2726, 6563, S27, 10), using
- discarding parts of the base [i] based on linear OA(8119, 6563, F81, 10) (dual of [6563, 6544, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(8117, 6561, F81, 9) (dual of [6561, 6544, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding parts of the base [i] based on linear OA(8119, 6563, F81, 10) (dual of [6563, 6544, 11]-code), using
- discarding factors based on OA(2726, 6563, S27, 10), using
- OA 5-folding and stacking [i] based on OA(2726, 6560, S27, 10), using
(16, 16+10, 2182)-Net over F27 — Digital
Digital (16, 26, 2182)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2726, 2182, F27, 10) (dual of [2182, 2156, 11]-code), using
- 1443 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 16 times 0, 1, 77 times 0, 1, 236 times 0, 1, 442 times 0, 1, 664 times 0) [i] based on linear OA(2719, 732, F27, 10) (dual of [732, 713, 11]-code), using
- construction XX applied to C1 = C([727,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([727,8]) [i] based on
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([727,8]) [i] based on
- 1443 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 16 times 0, 1, 77 times 0, 1, 236 times 0, 1, 442 times 0, 1, 664 times 0) [i] based on linear OA(2719, 732, F27, 10) (dual of [732, 713, 11]-code), using
(16, 16+10, 2779419)-Net in Base 27 — Upper bound on s
There is no (16, 26, 2779420)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 16 423211 957197 135738 127758 228829 171385 > 2726 [i]