Best Known (62−32, 62, s)-Nets in Base 27
(62−32, 62, 164)-Net over F27 — Constructive and digital
Digital (30, 62, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 23, 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 (7, 39, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 23, 82)-net over F27, using
(62−32, 62, 224)-Net in Base 27 — Constructive
(30, 62, 224)-net in base 27, using
- 6 times m-reduction [i] based on (30, 68, 224)-net in base 27, using
- base change [i] based on digital (13, 51, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 51, 224)-net over F81, using
(62−32, 62, 368)-Net over F27 — Digital
Digital (30, 62, 368)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2762, 368, F27, 2, 32) (dual of [(368, 2), 674, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2762, 736, F27, 32) (dual of [736, 674, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2762, 737, F27, 32) (dual of [737, 675, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- linear OA(2760, 729, F27, 32) (dual of [729, 669, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2754, 729, F27, 29) (dual of [729, 675, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2762, 737, F27, 32) (dual of [737, 675, 33]-code), using
- OOA 2-folding [i] based on linear OA(2762, 736, F27, 32) (dual of [736, 674, 33]-code), using
(62−32, 62, 92058)-Net in Base 27 — Upper bound on s
There is no (30, 62, 92059)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 55542 642802 331762 760928 694243 057005 070525 967204 130544 376333 628176 391490 814237 754723 355745 > 2762 [i]