Best Known (61, 61+23, s)-Nets in Base 27
(61, 61+23, 1865)-Net over F27 — Constructive and digital
Digital (61, 84, 1865)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (44, 67, 1789)-net over F27, using
- net defined by OOA [i] based on linear OOA(2767, 1789, F27, 23, 23) (dual of [(1789, 23), 41080, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2767, 19680, F27, 23) (dual of [19680, 19613, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2767, 19680, F27, 23) (dual of [19680, 19613, 24]-code), using
- net defined by OOA [i] based on linear OOA(2767, 1789, F27, 23, 23) (dual of [(1789, 23), 41080, 24]-NRT-code), using
- digital (6, 17, 76)-net over F27, using
(61, 61+23, 1890)-Net in Base 27 — Constructive
(61, 84, 1890)-net in base 27, using
- (u, u+v)-construction [i] based on
- (5, 16, 100)-net in base 27, using
- base change [i] based on digital (1, 12, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- base change [i] based on digital (1, 12, 100)-net over F81, using
- digital (45, 68, 1790)-net over F27, using
- net defined by OOA [i] based on linear OOA(2768, 1790, F27, 23, 23) (dual of [(1790, 23), 41102, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2768, 19691, F27, 23) (dual of [19691, 19623, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(2767, 19684, F27, 23) (dual of [19684, 19617, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2761, 19684, F27, 21) (dual of [19684, 19623, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 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([0,11]) ⊂ C([0,10]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(2768, 19691, F27, 23) (dual of [19691, 19623, 24]-code), using
- net defined by OOA [i] based on linear OOA(2768, 1790, F27, 23, 23) (dual of [(1790, 23), 41102, 24]-NRT-code), using
- (5, 16, 100)-net in base 27, using
(61, 61+23, 101656)-Net over F27 — Digital
Digital (61, 84, 101656)-net over F27, using
(61, 61+23, large)-Net in Base 27 — Upper bound on s
There is no (61, 84, large)-net in base 27, because
- 21 times m-reduction [i] would yield (61, 63, large)-net in base 27, but