Best Known (51−26, 51, s)-Nets in Base 128
(51−26, 51, 1260)-Net over F128 — Constructive and digital
Digital (25, 51, 1260)-net over F128, using
- net defined by OOA [i] based on linear OOA(12851, 1260, F128, 26, 26) (dual of [(1260, 26), 32709, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(12851, 16380, F128, 26) (dual of [16380, 16329, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(12851, 16380, F128, 26) (dual of [16380, 16329, 27]-code), using
(51−26, 51, 3545)-Net over F128 — Digital
Digital (25, 51, 3545)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12851, 3545, F128, 4, 26) (dual of [(3545, 4), 14129, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12851, 4096, F128, 4, 26) (dual of [(4096, 4), 16333, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- OOA 4-folding [i] based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(12851, 4096, F128, 4, 26) (dual of [(4096, 4), 16333, 27]-NRT-code), using
(51−26, 51, 8248005)-Net in Base 128 — Upper bound on s
There is no (25, 51, 8248006)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 293568 071428 805481 493525 820804 112160 265378 640656 964558 914456 737165 699708 985056 600970 200608 407523 791423 431912 > 12851 [i]