Best Known (15−8, 15, s)-Nets in Base 128
(15−8, 15, 4096)-Net over F128 — Constructive and digital
Digital (7, 15, 4096)-net over F128, using
- net defined by OOA [i] based on linear OOA(12815, 4096, F128, 8, 8) (dual of [(4096, 8), 32753, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on 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]
- OA 4-folding and stacking [i] based on linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using
(15−8, 15, 6175)-Net over F128 — Digital
Digital (7, 15, 6175)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12815, 6175, F128, 2, 8) (dual of [(6175, 2), 12335, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12815, 8193, F128, 2, 8) (dual of [(8193, 2), 16371, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12815, 16386, F128, 8) (dual of [16386, 16371, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(12815, 16386, F128, 8) (dual of [16386, 16371, 9]-code), using
- discarding factors / shortening the dual code based on linear OOA(12815, 8193, F128, 2, 8) (dual of [(8193, 2), 16371, 9]-NRT-code), using
(15−8, 15, 1390868)-Net in Base 128 — Upper bound on s
There is no (7, 15, 1390869)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 40 564905 188406 915756 623324 138413 > 12815 [i]