Best Known (29, 29+29, s)-Nets in Base 27
(29, 29+29, 166)-Net over F27 — Constructive and digital
Digital (29, 58, 166)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 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 (8, 37, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (7, 21, 82)-net over F27, using
(29, 29+29, 224)-Net in Base 27 — Constructive
(29, 58, 224)-net in base 27, using
- 6 times m-reduction [i] based on (29, 64, 224)-net in base 27, using
- base change [i] based on digital (13, 48, 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, 48, 224)-net over F81, using
(29, 29+29, 430)-Net over F27 — Digital
Digital (29, 58, 430)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2758, 430, F27, 29) (dual of [430, 372, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2758, 735, F27, 29) (dual of [735, 677, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- 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(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(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([0,14]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2758, 735, F27, 29) (dual of [735, 677, 30]-code), using
(29, 29+29, 156372)-Net in Base 27 — Upper bound on s
There is no (29, 58, 156373)-net in base 27, because
- 1 times m-reduction [i] would yield (29, 57, 156373)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 3870 409661 406141 089867 829016 461866 247378 657145 756241 262936 607673 053503 974654 810289 > 2757 [i]