Best Known (32, 32+30, s)-Nets in Base 27
(32, 32+30, 176)-Net over F27 — Constructive and digital
Digital (32, 62, 176)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 22, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (10, 40, 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 (7, 22, 82)-net over F27, using
(32, 32+30, 370)-Net in Base 27 — Constructive
(32, 62, 370)-net in base 27, using
- 2 times m-reduction [i] based on (32, 64, 370)-net in base 27, using
- base change [i] based on digital (16, 48, 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, 48, 370)-net over F81, using
(32, 32+30, 558)-Net over F27 — Digital
Digital (32, 62, 558)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2762, 558, F27, 30) (dual of [558, 496, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2762, 746, F27, 30) (dual of [746, 684, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- linear OA(2756, 729, F27, 30) (dual of [729, 673, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2745, 729, F27, 23) (dual of [729, 684, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(276, 17, F27, 6) (dual of [17, 11, 7]-code or 17-arc in PG(5,27)), using
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- Reed–Solomon code RS(21,27) [i]
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2762, 746, F27, 30) (dual of [746, 684, 31]-code), using
(32, 32+30, 203742)-Net in Base 27 — Upper bound on s
There is no (32, 62, 203743)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 55537 114693 313536 408093 980998 798660 214201 692815 969999 638391 645387 946578 404897 639692 634363 > 2762 [i]