Best Known (62−31, 62, s)-Nets in Base 27
(62−31, 62, 170)-Net over F27 — Constructive and digital
Digital (31, 62, 170)-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 (9, 40, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (7, 22, 82)-net over F27, using
(62−31, 62, 224)-Net in Base 27 — Constructive
(31, 62, 224)-net in base 27, using
- 10 times m-reduction [i] based on (31, 72, 224)-net in base 27, using
- base change [i] based on digital (13, 54, 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, 54, 224)-net over F81, using
(62−31, 62, 447)-Net over F27 — Digital
Digital (31, 62, 447)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2762, 447, F27, 31) (dual of [447, 385, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2762, 735, F27, 31) (dual of [735, 673, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- 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(2757, 730, F27, 29) (dual of [730, 673, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [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 C([0,15]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2762, 735, F27, 31) (dual of [735, 673, 32]-code), using
(62−31, 62, 163550)-Net in Base 27 — Upper bound on s
There is no (31, 62, 163551)-net in base 27, because
- 1 times m-reduction [i] would yield (31, 61, 163551)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 2056 839734 537754 272104 260433 363702 144906 706945 542349 408735 443692 972907 855752 020738 379003 > 2761 [i]