Best Known (41, 41+24, s)-Nets in Base 128
(41, 41+24, 1623)-Net over F128 — Constructive and digital
Digital (41, 65, 1623)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (6, 18, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (0, 12, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 6, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (23, 47, 1365)-net over F128, using
- net defined by OOA [i] based on linear OOA(12847, 1365, F128, 24, 24) (dual of [(1365, 24), 32713, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12847, 16380, F128, 24) (dual of [16380, 16333, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(12847, 16380, F128, 24) (dual of [16380, 16333, 25]-code), using
- net defined by OOA [i] based on linear OOA(12847, 1365, F128, 24, 24) (dual of [(1365, 24), 32713, 25]-NRT-code), using
- digital (6, 18, 258)-net over F128, using
(41, 41+24, 5463)-Net in Base 128 — Constructive
(41, 65, 5463)-net in base 128, using
- 1281 times duplication [i] based on (40, 64, 5463)-net in base 128, using
- t-expansion [i] based on (39, 64, 5463)-net in base 128, using
- base change [i] based on digital (31, 56, 5463)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 5463, F256, 25, 25) (dual of [(5463, 25), 136519, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25656, 65557, F256, 25) (dual of [65557, 65501, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65560, F256, 25) (dual of [65560, 65504, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2567, 23, F256, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,256)), using
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- Reed–Solomon code RS(249,256) [i]
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25656, 65560, F256, 25) (dual of [65560, 65504, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25656, 65557, F256, 25) (dual of [65557, 65501, 26]-code), using
- net defined by OOA [i] based on linear OOA(25656, 5463, F256, 25, 25) (dual of [(5463, 25), 136519, 26]-NRT-code), using
- base change [i] based on digital (31, 56, 5463)-net over F256, using
- t-expansion [i] based on (39, 64, 5463)-net in base 128, using
(41, 41+24, 66971)-Net over F128 — Digital
Digital (41, 65, 66971)-net over F128, using
(41, 41+24, large)-Net in Base 128 — Upper bound on s
There is no (41, 65, large)-net in base 128, because
- 22 times m-reduction [i] would yield (41, 43, large)-net in base 128, but