Best Known (18, 33, s)-Nets in Base 27
(18, 33, 152)-Net over F27 — Constructive and digital
Digital (18, 33, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 76)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 730)-net over F27, using
- net defined by OOA [i] based on linear OOA(274, 730, F27, 3, 3) (dual of [(730, 3), 2186, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(274, 730, F27, 2, 3) (dual of [(730, 2), 1456, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(274, 730, F27, 3, 3) (dual of [(730, 3), 2186, 4]-NRT-code), using
- digital (1, 8, 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, 4, 730)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 21, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (5, 12, 76)-net over F27, using
(18, 33, 200)-Net in Base 27 — Constructive
(18, 33, 200)-net in base 27, using
- 271 times duplication [i] based on (17, 32, 200)-net in base 27, using
- base change [i] based on digital (9, 24, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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, 16, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 8, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (9, 24, 200)-net over F81, using
(18, 33, 722)-Net over F27 — Digital
Digital (18, 33, 722)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2733, 722, F27, 15) (dual of [722, 689, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2733, 743, F27, 15) (dual of [743, 710, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(2729, 729, F27, 15) (dual of [729, 700, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(274, 14, F27, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(2733, 743, F27, 15) (dual of [743, 710, 16]-code), using
(18, 33, 454275)-Net in Base 27 — Upper bound on s
There is no (18, 33, 454276)-net in base 27, because
- 1 times m-reduction [i] would yield (18, 32, 454276)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 6362 727571 529336 148776 415501 396419 017282 862673 > 2732 [i]