Best Known (17, 17+11, s)-Nets in Base 128
(17, 17+11, 6554)-Net over F128 — Constructive and digital
Digital (17, 28, 6554)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- appending kth column [i] based on linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
- digital (10, 21, 3277)-net over F128, using
- net defined by OOA [i] based on linear OOA(12821, 3277, F128, 11, 11) (dual of [(3277, 11), 36026, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- net defined by OOA [i] based on linear OOA(12821, 3277, F128, 11, 11) (dual of [(3277, 11), 36026, 12]-NRT-code), using
- digital (2, 7, 8128)-net over F128, using
(17, 17+11, 13109)-Net in Base 128 — Constructive
(17, 28, 13109)-net in base 128, using
- net defined by OOA [i] based on OOA(12828, 13109, S128, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12828, 65546, S128, 11), using
- discarding factors based on OA(12828, 65548, S128, 11), using
- discarding parts of the base [i] based on linear OA(25624, 65548, F256, 11) (dual of [65548, 65524, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
- discarding parts of the base [i] based on linear OA(25624, 65548, F256, 11) (dual of [65548, 65524, 12]-code), using
- discarding factors based on OA(12828, 65548, S128, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12828, 65546, S128, 11), using
(17, 17+11, 32643)-Net over F128 — Digital
Digital (17, 28, 32643)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12828, 32643, F128, 11) (dual of [32643, 32615, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- (u, u+v)-construction [i] based on
(17, 17+11, large)-Net in Base 128 — Upper bound on s
There is no (17, 28, large)-net in base 128, because
- 9 times m-reduction [i] would yield (17, 19, large)-net in base 128, but