Best Known (10, 20, s)-Nets in Base 128
(10, 20, 3277)-Net over F128 — Constructive and digital
Digital (10, 20, 3277)-net over F128, using
- 1 times m-reduction [i] based on 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
(10, 20, 6974)-Net over F128 — Digital
Digital (10, 20, 6974)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12820, 6974, F128, 2, 10) (dual of [(6974, 2), 13928, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12820, 8194, F128, 2, 10) (dual of [(8194, 2), 16368, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12820, 16388, F128, 10) (dual of [16388, 16368, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12820, 16389, F128, 10) (dual of [16389, 16369, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(12820, 16389, F128, 10) (dual of [16389, 16369, 11]-code), using
- OOA 2-folding [i] based on linear OA(12820, 16388, F128, 10) (dual of [16388, 16368, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(12820, 8194, F128, 2, 10) (dual of [(8194, 2), 16368, 11]-NRT-code), using
(10, 20, 5506456)-Net in Base 128 — Upper bound on s
There is no (10, 20, 5506457)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 1 393796 673776 826851 621939 197487 553557 095800 > 12820 [i]