Best Known (39−20, 39, s)-Nets in Base 128
(39−20, 39, 1638)-Net over F128 — Constructive and digital
Digital (19, 39, 1638)-net over F128, using
- net defined by OOA [i] based on linear OOA(12839, 1638, F128, 20, 20) (dual of [(1638, 20), 32721, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12839, 16380, F128, 20) (dual of [16380, 16341, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12839, 16380, F128, 20) (dual of [16380, 16341, 21]-code), using
(39−20, 39, 4096)-Net over F128 — Digital
Digital (19, 39, 4096)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12839, 4096, F128, 4, 20) (dual of [(4096, 4), 16345, 21]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- OOA 4-folding [i] based on linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using
(39−20, 39, 5892385)-Net in Base 128 — Upper bound on s
There is no (19, 39, 5892386)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 15177 122560 796841 112115 628325 690341 766152 085932 876791 083484 537293 497746 608877 870352 > 12839 [i]