Best Known (12, 12+13, s)-Nets in Base 128
(12, 12+13, 2730)-Net over F128 — Constructive and digital
Digital (12, 25, 2730)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 2730, F128, 13, 13) (dual of [(2730, 13), 35465, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12825, 16381, F128, 13) (dual of [16381, 16356, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12825, 16381, F128, 13) (dual of [16381, 16356, 14]-code), using
(12, 12+13, 4619)-Net over F128 — Digital
Digital (12, 25, 4619)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12825, 4619, F128, 3, 13) (dual of [(4619, 3), 13832, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12825, 5462, F128, 3, 13) (dual of [(5462, 3), 16361, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12825, 16386, F128, 13) (dual of [16386, 16361, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- OOA 3-folding [i] based on linear OA(12825, 16386, F128, 13) (dual of [16386, 16361, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(12825, 5462, F128, 3, 13) (dual of [(5462, 3), 16361, 14]-NRT-code), using
(12, 12+13, 6327877)-Net in Base 128 — Upper bound on s
There is no (12, 25, 6327878)-net in base 128, because
- 1 times m-reduction [i] would yield (12, 24, 6327878)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 374 144584 166625 438267 789511 715508 227626 240305 056096 > 12824 [i]