Best Known (72−35, 72, s)-Nets in Base 27
(72−35, 72, 188)-Net over F27 — Constructive and digital
Digital (37, 72, 188)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 27, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (10, 45, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27 (see above)
- digital (10, 27, 94)-net over F27, using
(72−35, 72, 370)-Net in Base 27 — Constructive
(37, 72, 370)-net in base 27, using
- 12 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
(72−35, 72, 593)-Net over F27 — Digital
Digital (37, 72, 593)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2772, 593, F27, 35) (dual of [593, 521, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2772, 741, F27, 35) (dual of [741, 669, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,15]) [i] based on
- linear OA(2769, 730, F27, 35) (dual of [730, 661, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(2761, 730, F27, 31) (dual of [730, 669, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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([0,17]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2772, 741, F27, 35) (dual of [741, 669, 36]-code), using
(72−35, 72, 262425)-Net in Base 27 — Upper bound on s
There is no (37, 72, 262426)-net in base 27, because
- 1 times m-reduction [i] would yield (37, 71, 262426)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 423485 448973 544895 020214 358674 808543 505229 911253 032124 584930 419488 076166 776730 604508 052237 864026 162437 > 2771 [i]