Best Known (42, 42+24, s)-Nets in Base 27
(42, 42+24, 256)-Net over F27 — Constructive and digital
Digital (42, 66, 256)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 6, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- digital (2, 8, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27 (see above)
- digital (2, 10, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27 (see above)
- digital (2, 14, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27 (see above)
- digital (4, 28, 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 (2, 6, 48)-net over F27, using
(42, 42+24, 547)-Net in Base 27 — Constructive
(42, 66, 547)-net in base 27, using
- 1 times m-reduction [i] based on (42, 67, 547)-net in base 27, using
- net defined by OOA [i] based on OOA(2767, 547, S27, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(2767, 6565, S27, 25), using
- discarding factors based on OA(2767, 6567, S27, 25), using
- discarding parts of the base [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- 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(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,12]) ⊂ C([0,11]) [i] based on
- discarding parts of the base [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- discarding factors based on OA(2767, 6567, S27, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(2767, 6565, S27, 25), using
- net defined by OOA [i] based on OOA(2767, 547, S27, 25, 25), using
(42, 42+24, 4655)-Net over F27 — Digital
Digital (42, 66, 4655)-net over F27, using
(42, 42+24, large)-Net in Base 27 — Upper bound on s
There is no (42, 66, large)-net in base 27, because
- 22 times m-reduction [i] would yield (42, 44, large)-net in base 27, but