Best Known (31, 31+19, s)-Nets in Base 128
(31, 31+19, 2078)-Net over F128 — Constructive and digital
Digital (31, 50, 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 (18, 37, 1820)-net over F128, using
- net defined by OOA [i] based on linear OOA(12837, 1820, F128, 19, 19) (dual of [(1820, 19), 34543, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12837, 16381, F128, 19) (dual of [16381, 16344, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12837, 16381, F128, 19) (dual of [16381, 16344, 20]-code), using
- net defined by OOA [i] based on linear OOA(12837, 1820, F128, 19, 19) (dual of [(1820, 19), 34543, 20]-NRT-code), using
- digital (4, 13, 258)-net over F128, using
(31, 31+19, 7283)-Net in Base 128 — Constructive
(31, 50, 7283)-net in base 128, using
- 1282 times duplication [i] based on (29, 48, 7283)-net in base 128, using
- base change [i] based on digital (23, 42, 7283)-net over F256, using
- 2562 times duplication [i] based on digital (21, 40, 7283)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
- 2562 times duplication [i] based on digital (21, 40, 7283)-net over F256, using
- base change [i] based on digital (23, 42, 7283)-net over F256, using
(31, 31+19, 42440)-Net over F128 — Digital
Digital (31, 50, 42440)-net over F128, using
(31, 31+19, large)-Net in Base 128 — Upper bound on s
There is no (31, 50, large)-net in base 128, because
- 17 times m-reduction [i] would yield (31, 33, large)-net in base 128, but