Best Known (53−20, 53, s)-Nets in Base 128
(53−20, 53, 1830)-Net over F128 — Constructive and digital
Digital (33, 53, 1830)-net over F128, using
- 1 times m-reduction [i] based on digital (33, 54, 1830)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (20, 41, 1638)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- digital (3, 13, 192)-net over F128, using
- (u, u+v)-construction [i] based on
(53−20, 53, 6555)-Net in Base 128 — Constructive
(33, 53, 6555)-net in base 128, using
- t-expansion [i] based on (32, 53, 6555)-net in base 128, using
- net defined by OOA [i] based on OOA(12853, 6555, S128, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12853, 65551, S128, 21), using
- discarding factors based on OA(12853, 65554, S128, 21), using
- discarding parts of the base [i] based on linear OA(25646, 65554, F256, 21) (dual of [65554, 65508, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(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([0,10]) ⊂ C([0,7]) [i] based on
- discarding parts of the base [i] based on linear OA(25646, 65554, F256, 21) (dual of [65554, 65508, 22]-code), using
- discarding factors based on OA(12853, 65554, S128, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12853, 65551, S128, 21), using
- net defined by OOA [i] based on OOA(12853, 6555, S128, 21, 21), using
(53−20, 53, 47153)-Net over F128 — Digital
Digital (33, 53, 47153)-net over F128, using
(53−20, 53, large)-Net in Base 128 — Upper bound on s
There is no (33, 53, large)-net in base 128, because
- 18 times m-reduction [i] would yield (33, 35, large)-net in base 128, but