Best Known (45, 45+31, s)-Nets in Base 128
(45, 45+31, 1221)-Net over F128 — Constructive and digital
Digital (45, 76, 1221)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 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 (30, 61, 1092)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
- digital (0, 15, 129)-net over F128, using
(45, 45+31, 4370)-Net in Base 128 — Constructive
(45, 76, 4370)-net in base 128, using
- net defined by OOA [i] based on OOA(12876, 4370, S128, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(12876, 65551, S128, 31), using
- discarding factors based on OA(12876, 65554, S128, 31), using
- discarding parts of the base [i] based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding parts of the base [i] based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- discarding factors based on OA(12876, 65554, S128, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(12876, 65551, S128, 31), using
(45, 45+31, 20681)-Net over F128 — Digital
Digital (45, 76, 20681)-net over F128, using
(45, 45+31, large)-Net in Base 128 — Upper bound on s
There is no (45, 76, large)-net in base 128, because
- 29 times m-reduction [i] would yield (45, 47, large)-net in base 128, but