Best Known (19, 19+13, s)-Nets in Base 128
(19, 19+13, 2880)-Net over F128 — Constructive and digital
Digital (19, 32, 2880)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (12, 25, 2730)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 2730, F128, 13, 13) (dual of [(2730, 13), 35465, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12825, 16381, F128, 13) (dual of [16381, 16356, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12825, 16381, F128, 13) (dual of [16381, 16356, 14]-code), using
- net defined by OOA [i] based on linear OOA(12825, 2730, F128, 13, 13) (dual of [(2730, 13), 35465, 14]-NRT-code), using
- digital (1, 7, 150)-net over F128, using
(19, 19+13, 10924)-Net in Base 128 — Constructive
(19, 32, 10924)-net in base 128, using
- base change [i] based on digital (15, 28, 10924)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 10924, F256, 13, 13) (dual of [(10924, 13), 141984, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25628, 65545, F256, 13) (dual of [65545, 65517, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25628, 65545, F256, 13) (dual of [65545, 65517, 14]-code), using
- net defined by OOA [i] based on linear OOA(25628, 10924, F256, 13, 13) (dual of [(10924, 13), 141984, 14]-NRT-code), using
(19, 19+13, 17335)-Net over F128 — Digital
Digital (19, 32, 17335)-net over F128, using
(19, 19+13, 32419)-Net in Base 128
(19, 32, 32419)-net in base 128, using
- base change [i] based on digital (15, 28, 32419)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 32419, F256, 2, 13) (dual of [(32419, 2), 64810, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 32774, F256, 2, 13) (dual of [(32774, 2), 65520, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
- OOA 2-folding [i] based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 32774, F256, 2, 13) (dual of [(32774, 2), 65520, 14]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 32419, F256, 2, 13) (dual of [(32419, 2), 64810, 14]-NRT-code), using
(19, 19+13, large)-Net in Base 128 — Upper bound on s
There is no (19, 32, large)-net in base 128, because
- 11 times m-reduction [i] would yield (19, 21, large)-net in base 128, but