Best Known (20−9, 20, s)-Nets in Base 27
(20−9, 20, 185)-Net over F27 — Constructive and digital
Digital (11, 20, 185)-net over F27, using
- net defined by OOA [i] based on linear OOA(2720, 185, F27, 9, 9) (dual of [(185, 9), 1645, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2720, 741, F27, 9) (dual of [741, 721, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(2717, 730, F27, 9) (dual of [730, 713, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(279, 730, F27, 5) (dual of [730, 721, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(273, 11, F27, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(2720, 741, F27, 9) (dual of [741, 721, 10]-code), using
(20−9, 20, 200)-Net in Base 27 — Constructive
(11, 20, 200)-net in base 27, using
- base change [i] based on digital (6, 15, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (1, 10, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 5, 100)-net over F81, using
- (u, u+v)-construction [i] based on
(20−9, 20, 774)-Net over F27 — Digital
Digital (11, 20, 774)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2720, 774, F27, 9) (dual of [774, 754, 10]-code), using
- 39 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0) [i] based on linear OA(2717, 732, F27, 9) (dual of [732, 715, 10]-code), using
- construction XX applied to C1 = C([727,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([727,7]) [i] based on
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [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(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(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [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([727,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([727,7]) [i] based on
- 39 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0) [i] based on linear OA(2717, 732, F27, 9) (dual of [732, 715, 10]-code), using
(20−9, 20, 535865)-Net in Base 27 — Upper bound on s
There is no (11, 20, 535866)-net in base 27, because
- 1 times m-reduction [i] would yield (11, 19, 535866)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 1570 045800 582659 481661 476521 > 2719 [i]