Best Known (72−27, 72, s)-Nets in Base 27
(72−27, 72, 260)-Net over F27 — Constructive and digital
Digital (45, 72, 260)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 10, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (4, 13, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (4, 17, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (5, 32, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 10, 64)-net over F27, using
(72−27, 72, 505)-Net in Base 27 — Constructive
(45, 72, 505)-net in base 27, using
- base change [i] based on digital (27, 54, 505)-net over F81, using
- net defined by OOA [i] based on linear OOA(8154, 505, F81, 27, 27) (dual of [(505, 27), 13581, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8154, 6566, F81, 27) (dual of [6566, 6512, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8154, 6567, F81, 27) (dual of [6567, 6513, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 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(8154, 6567, F81, 27) (dual of [6567, 6513, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8154, 6566, F81, 27) (dual of [6566, 6512, 28]-code), using
- net defined by OOA [i] based on linear OOA(8154, 505, F81, 27, 27) (dual of [(505, 27), 13581, 28]-NRT-code), using
(72−27, 72, 3746)-Net over F27 — Digital
Digital (45, 72, 3746)-net over F27, using
(72−27, 72, large)-Net in Base 27 — Upper bound on s
There is no (45, 72, large)-net in base 27, because
- 25 times m-reduction [i] would yield (45, 47, large)-net in base 27, but