Best Known (17, 28, s)-Nets in Base 27
(17, 28, 292)-Net over F27 — Constructive and digital
Digital (17, 28, 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 (10, 21, 146)-net over F27, using
- net defined by OOA [i] based on linear OOA(2721, 146, F27, 11, 11) (dual of [(146, 11), 1585, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2721, 731, F27, 11) (dual of [731, 710, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(2721, 729, F27, 11) (dual of [729, 708, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(2721, 731, F27, 11) (dual of [731, 710, 12]-code), using
- net defined by OOA [i] based on linear OOA(2721, 146, F27, 11, 11) (dual of [(146, 11), 1585, 12]-NRT-code), using
- digital (2, 7, 351)-net over F27, using
(17, 28, 1312)-Net in Base 27 — Constructive
(17, 28, 1312)-net in base 27, using
- base change [i] based on digital (10, 21, 1312)-net over F81, using
- net defined by OOA [i] based on linear OOA(8121, 1312, F81, 11, 11) (dual of [(1312, 11), 14411, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 5-folding and stacking with additional row [i] based on linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using
- net defined by OOA [i] based on linear OOA(8121, 1312, F81, 11, 11) (dual of [(1312, 11), 14411, 12]-NRT-code), using
(17, 28, 1779)-Net over F27 — Digital
Digital (17, 28, 1779)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2728, 1779, F27, 11) (dual of [1779, 1751, 12]-code), using
- 1040 step Varšamov–Edel lengthening with (ri) = (2, 1, 8 times 0, 1, 45 times 0, 1, 160 times 0, 1, 328 times 0, 1, 493 times 0) [i] based on linear OA(2721, 732, F27, 11) (dual of [732, 711, 12]-code), using
- construction XX applied to C1 = C([727,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([727,9]) [i] based on
- 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(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [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(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([727,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([727,9]) [i] based on
- 1040 step Varšamov–Edel lengthening with (ri) = (2, 1, 8 times 0, 1, 45 times 0, 1, 160 times 0, 1, 328 times 0, 1, 493 times 0) [i] based on linear OA(2721, 732, F27, 11) (dual of [732, 711, 12]-code), using
(17, 28, 2187)-Net in Base 27
(17, 28, 2187)-net in base 27, using
- base change [i] based on digital (10, 21, 2187)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8121, 2187, F81, 3, 11) (dual of [(2187, 3), 6540, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 3-folding [i] based on linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8121, 2187, F81, 3, 11) (dual of [(2187, 3), 6540, 12]-NRT-code), using
(17, 28, 5373126)-Net in Base 27 — Upper bound on s
There is no (17, 28, 5373127)-net in base 27, because
- 1 times m-reduction [i] would yield (17, 27, 5373127)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 443 426770 237067 339127 518503 587118 121487 > 2727 [i]