Best Known (27, 54, s)-Nets in Base 27
(27, 54, 164)-Net over F27 — Constructive and digital
Digital (27, 54, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 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, 34, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 20, 82)-net over F27, using
(27, 54, 224)-Net in Base 27 — Constructive
(27, 54, 224)-net in base 27, using
- 2 times m-reduction [i] based on (27, 56, 224)-net in base 27, using
- base change [i] based on digital (13, 42, 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, 42, 224)-net over F81, using
(27, 54, 413)-Net over F27 — Digital
Digital (27, 54, 413)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2754, 413, F27, 27) (dual of [413, 359, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2754, 735, F27, 27) (dual of [735, 681, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(2753, 730, F27, 27) (dual of [730, 677, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(2749, 730, F27, 25) (dual of [730, 681, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2754, 735, F27, 27) (dual of [735, 681, 28]-code), using
(27, 54, 149270)-Net in Base 27 — Upper bound on s
There is no (27, 54, 149271)-net in base 27, because
- 1 times m-reduction [i] would yield (27, 53, 149271)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 7282 823688 096223 645211 112513 162601 243853 663220 418102 277527 619121 487029 930463 > 2753 [i]