Best Known (12, 12+12, s)-Nets in Base 128
(12, 12+12, 2731)-Net over F128 — Constructive and digital
Digital (12, 24, 2731)-net over F128, using
- 1281 times duplication [i] based on digital (11, 23, 2731)-net over F128, using
- net defined by OOA [i] based on linear OOA(12823, 2731, F128, 12, 12) (dual of [(2731, 12), 32749, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(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(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(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- net defined by OOA [i] based on linear OOA(12823, 2731, F128, 12, 12) (dual of [(2731, 12), 32749, 13]-NRT-code), using
(12, 12+12, 5463)-Net over F128 — Digital
Digital (12, 24, 5463)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12824, 5463, F128, 3, 12) (dual of [(5463, 3), 16365, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12824, 16389, F128, 12) (dual of [16389, 16365, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(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(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(11) ⊂ Ce(9) [i] based on
- OOA 3-folding [i] based on linear OA(12824, 16389, F128, 12) (dual of [16389, 16365, 13]-code), using
(12, 12+12, 6327877)-Net in Base 128 — Upper bound on s
There is no (12, 24, 6327878)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 374 144584 166625 438267 789511 715508 227626 240305 056096 > 12824 [i]