Best Known (18−7, 18, s)-Nets in Base 27
(18−7, 18, 776)-Net over F27 — Constructive and digital
Digital (11, 18, 776)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 28)-net over F27, using
- s-reduction based on digital (0, 0, s)-net over F27 with arbitrarily large s, using
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (1, 4, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- digital (1, 8, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27 (see above)
- digital (0, 0, 28)-net over F27, using
(18−7, 18, 2187)-Net in Base 27 — Constructive
(11, 18, 2187)-net in base 27, using
- net defined by OOA [i] based on OOA(2718, 2187, S27, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(2718, 6562, S27, 7), using
- discarding factors based on OA(2718, 6563, S27, 7), using
- discarding parts of the base [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
- discarding factors based on OA(2718, 6563, S27, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(2718, 6562, S27, 7), using
(18−7, 18, 2282)-Net over F27 — Digital
Digital (11, 18, 2282)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2718, 2282, F27, 7) (dual of [2282, 2264, 8]-code), using
- 819 step Varšamov–Edel lengthening with (ri) = (1, 818 times 0) [i] based on linear OA(2717, 1462, F27, 7) (dual of [1462, 1445, 8]-code), using
- (u, u+v)-construction [i] based on
- linear OA(274, 730, F27, 3) (dual of [730, 726, 4]-code or 730-cap in PG(3,27)), using
- 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
- (u, u+v)-construction [i] based on
- 819 step Varšamov–Edel lengthening with (ri) = (1, 818 times 0) [i] based on linear OA(2717, 1462, F27, 7) (dual of [1462, 1445, 8]-code), using
(18−7, 18, large)-Net in Base 27 — Upper bound on s
There is no (11, 18, large)-net in base 27, because
- 5 times m-reduction [i] would yield (11, 13, large)-net in base 27, but