Best Known (30, 63, s)-Nets in Base 27
(30, 63, 164)-Net over F27 — Constructive and digital
Digital (30, 63, 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, 40, 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
(30, 63, 224)-Net in Base 27 — Constructive
(30, 63, 224)-net in base 27, using
- 5 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
(30, 63, 362)-Net over F27 — Digital
Digital (30, 63, 362)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2763, 362, F27, 2, 33) (dual of [(362, 2), 661, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2763, 367, F27, 2, 33) (dual of [(367, 2), 671, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2763, 734, F27, 33) (dual of [734, 671, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2762, 729, F27, 33) (dual of [729, 667, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2758, 729, F27, 31) (dual of [729, 671, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(2763, 734, F27, 33) (dual of [734, 671, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2763, 367, F27, 2, 33) (dual of [(367, 2), 671, 34]-NRT-code), using
(30, 63, 92058)-Net in Base 27 — Upper bound on s
There is no (30, 63, 92059)-net in base 27, because
- 1 times m-reduction [i] would yield (30, 62, 92059)-net in base 27, but
- 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]