Best Known (56−21, 56, s)-Nets in Base 128
(56−21, 56, 1896)-Net over F128 — Constructive and digital
Digital (35, 56, 1896)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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, 10, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 5, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (20, 41, 1638)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- digital (5, 15, 258)-net over F128, using
(56−21, 56, 6556)-Net in Base 128 — Constructive
(35, 56, 6556)-net in base 128, using
- base change [i] based on digital (28, 49, 6556)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 6556, F256, 21, 21) (dual of [(6556, 21), 137627, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25649, 65561, F256, 21) (dual of [65561, 65512, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, 65562, F256, 21) (dual of [65562, 65513, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(11) [i] based on
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2568, 26, F256, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,256)), using
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- Reed–Solomon code RS(248,256) [i]
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- construction X applied to Ce(20) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(25649, 65562, F256, 21) (dual of [65562, 65513, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25649, 65561, F256, 21) (dual of [65561, 65512, 22]-code), using
- net defined by OOA [i] based on linear OOA(25649, 6556, F256, 21, 21) (dual of [(6556, 21), 137627, 22]-NRT-code), using
(56−21, 56, 51973)-Net over F128 — Digital
Digital (35, 56, 51973)-net over F128, using
(56−21, 56, large)-Net in Base 128 — Upper bound on s
There is no (35, 56, large)-net in base 128, because
- 19 times m-reduction [i] would yield (35, 37, large)-net in base 128, but