Best Known (48−18, 48, s)-Nets in Base 128
(48−18, 48, 2078)-Net over F128 — Constructive and digital
Digital (30, 48, 2078)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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, 9, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 4, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (17, 35, 1820)-net over F128, using
- net defined by OOA [i] based on linear OOA(12835, 1820, F128, 18, 18) (dual of [(1820, 18), 32725, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12835, 16380, F128, 18) (dual of [16380, 16345, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(12835, 16380, F128, 18) (dual of [16380, 16345, 19]-code), using
- net defined by OOA [i] based on linear OOA(12835, 1820, F128, 18, 18) (dual of [(1820, 18), 32725, 19]-NRT-code), using
- digital (4, 13, 258)-net over F128, using
(48−18, 48, 7284)-Net in Base 128 — Constructive
(30, 48, 7284)-net in base 128, using
- base change [i] based on digital (24, 42, 7284)-net over F256, using
- 2561 times duplication [i] based on digital (23, 41, 7284)-net over F256, using
- net defined by OOA [i] based on linear OOA(25641, 7284, F256, 18, 18) (dual of [(7284, 18), 131071, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(25641, 65556, F256, 18) (dual of [65556, 65515, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- 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(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2566, 20, F256, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,256)), using
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- Reed–Solomon code RS(250,256) [i]
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- OA 9-folding and stacking [i] based on linear OA(25641, 65556, F256, 18) (dual of [65556, 65515, 19]-code), using
- net defined by OOA [i] based on linear OOA(25641, 7284, F256, 18, 18) (dual of [(7284, 18), 131071, 19]-NRT-code), using
- 2561 times duplication [i] based on digital (23, 41, 7284)-net over F256, using
(48−18, 48, 50348)-Net over F128 — Digital
Digital (30, 48, 50348)-net over F128, using
(48−18, 48, large)-Net in Base 128 — Upper bound on s
There is no (30, 48, large)-net in base 128, because
- 16 times m-reduction [i] would yield (30, 32, large)-net in base 128, but