Best Known (9, 9+10, s)-Nets in Base 128
(9, 9+10, 3277)-Net over F128 — Constructive and digital
Digital (9, 19, 3277)-net over F128, using
- net defined by OOA [i] based on linear OOA(12819, 3277, F128, 10, 10) (dual of [(3277, 10), 32751, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12819, 16385, F128, 10) (dual of [16385, 16366, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [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(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(12819, 16385, F128, 10) (dual of [16385, 16366, 11]-code), using
(9, 9+10, 5462)-Net over F128 — Digital
Digital (9, 19, 5462)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12819, 5462, F128, 3, 10) (dual of [(5462, 3), 16367, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [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(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- OOA 3-folding [i] based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
(9, 9+10, 2086555)-Net in Base 128 — Upper bound on s
There is no (9, 19, 2086556)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 10889 042832 991671 701373 147708 865919 888070 > 12819 [i]