Best Known (57−23, 57, s)-Nets in Base 128
(57−23, 57, 1639)-Net over F128 — Constructive and digital
Digital (34, 57, 1639)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (22, 45, 1489)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- digital (1, 12, 150)-net over F128, using
(57−23, 57, 5959)-Net in Base 128 — Constructive
(34, 57, 5959)-net in base 128, using
- 1281 times duplication [i] based on (33, 56, 5959)-net in base 128, using
- base change [i] based on digital (26, 49, 5959)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5959, F256, 23, 23) (dual of [(5959, 23), 137008, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(25649, 65550, F256, 23) (dual of [65550, 65501, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(25649, 65550, F256, 23) (dual of [65550, 65501, 24]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5959, F256, 23, 23) (dual of [(5959, 23), 137008, 24]-NRT-code), using
- base change [i] based on digital (26, 49, 5959)-net over F256, using
(57−23, 57, 20553)-Net over F128 — Digital
Digital (34, 57, 20553)-net over F128, using
(57−23, 57, 21037)-Net in Base 128
(34, 57, 21037)-net in base 128, using
- 1281 times duplication [i] based on (33, 56, 21037)-net in base 128, using
- base change [i] based on digital (26, 49, 21037)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 21037, F256, 3, 23) (dual of [(21037, 3), 63062, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 21850, F256, 3, 23) (dual of [(21850, 3), 65501, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25649, 65550, F256, 23) (dual of [65550, 65501, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- OOA 3-folding [i] based on linear OA(25649, 65550, F256, 23) (dual of [65550, 65501, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 21850, F256, 3, 23) (dual of [(21850, 3), 65501, 24]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 21037, F256, 3, 23) (dual of [(21037, 3), 63062, 24]-NRT-code), using
- base change [i] based on digital (26, 49, 21037)-net over F256, using
(57−23, 57, large)-Net in Base 128 — Upper bound on s
There is no (34, 57, large)-net in base 128, because
- 21 times m-reduction [i] would yield (34, 36, large)-net in base 128, but