Best Known (15−7, 15, s)-Nets in Base 27
(15−7, 15, 245)-Net over F27 — Constructive and digital
Digital (8, 15, 245)-net over F27, using
- net defined by OOA [i] based on linear OOA(2715, 245, F27, 7, 7) (dual of [(245, 7), 1700, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2715, 736, F27, 7) (dual of [736, 721, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2715, 737, F27, 7) (dual of [737, 722, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2713, 729, F27, 7) (dual of [729, 716, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(277, 729, F27, 4) (dual of [729, 722, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2715, 737, F27, 7) (dual of [737, 722, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2715, 736, F27, 7) (dual of [736, 721, 8]-code), using
(15−7, 15, 753)-Net over F27 — Digital
Digital (8, 15, 753)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2715, 753, F27, 7) (dual of [753, 738, 8]-code), using
- 19 step Varšamov–Edel lengthening with (ri) = (2, 18 times 0) [i] based on linear OA(2713, 732, F27, 7) (dual of [732, 719, 8]-code), using
- construction XX applied to C1 = C([727,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([727,5]) [i] based on
- linear OA(2711, 728, F27, 6) (dual of [728, 717, 7]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2711, 728, F27, 6) (dual of [728, 717, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(279, 728, F27, 5) (dual of [728, 719, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([727,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([727,5]) [i] based on
- 19 step Varšamov–Edel lengthening with (ri) = (2, 18 times 0) [i] based on linear OA(2713, 732, F27, 7) (dual of [732, 719, 8]-code), using
(15−7, 15, 334277)-Net in Base 27 — Upper bound on s
There is no (8, 15, 334278)-net in base 27, because
- 1 times m-reduction [i] would yield (8, 14, 334278)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 109 419953 566496 048113 > 2714 [i]