Best Known (13, 13+13, s)-Nets in Base 27
(13, 13+13, 122)-Net over F27 — Constructive and digital
Digital (13, 26, 122)-net over F27, using
- net defined by OOA [i] based on linear OOA(2726, 122, F27, 13, 13) (dual of [(122, 13), 1560, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2726, 733, F27, 13) (dual of [733, 707, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2726, 735, F27, 13) (dual of [735, 709, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(2725, 730, F27, 13) (dual of [730, 705, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2721, 730, F27, 11) (dual of [730, 709, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2726, 735, F27, 13) (dual of [735, 709, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2726, 733, F27, 13) (dual of [733, 707, 14]-code), using
(13, 13+13, 164)-Net in Base 27 — Constructive
(13, 26, 164)-net in base 27, using
- (u, u+v)-construction [i] based on
- (2, 8, 82)-net in base 27, using
- base change [i] based on digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base change [i] based on digital (0, 6, 82)-net over F81, using
- (5, 18, 82)-net in base 27, using
- 2 times m-reduction [i] based on (5, 20, 82)-net in base 27, using
- base change [i] based on digital (0, 15, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- base change [i] based on digital (0, 15, 82)-net over F81, using
- 2 times m-reduction [i] based on (5, 20, 82)-net in base 27, using
- (2, 8, 82)-net in base 27, using
(13, 13+13, 367)-Net over F27 — Digital
Digital (13, 26, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2726, 367, F27, 2, 13) (dual of [(367, 2), 708, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2726, 734, F27, 13) (dual of [734, 708, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2726, 735, F27, 13) (dual of [735, 709, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(2725, 730, F27, 13) (dual of [730, 705, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2721, 730, F27, 11) (dual of [730, 709, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2726, 735, F27, 13) (dual of [735, 709, 14]-code), using
- OOA 2-folding [i] based on linear OA(2726, 734, F27, 13) (dual of [734, 708, 14]-code), using
(13, 13+13, 105987)-Net in Base 27 — Upper bound on s
There is no (13, 26, 105988)-net in base 27, because
- 1 times m-reduction [i] would yield (13, 25, 105988)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 608295 928706 555952 726926 847727 418841 > 2725 [i]