Best Known (27, 51, s)-Nets in Base 128
(27, 51, 1366)-Net over F128 — Constructive and digital
Digital (27, 51, 1366)-net over F128, using
- 1 times m-reduction [i] based on digital (27, 52, 1366)-net over F128, using
- net defined by OOA [i] based on linear OOA(12852, 1366, F128, 25, 25) (dual of [(1366, 25), 34098, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12852, 16393, F128, 25) (dual of [16393, 16341, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12852, 16396, F128, 25) (dual of [16396, 16344, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12852, 16396, F128, 25) (dual of [16396, 16344, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12852, 16393, F128, 25) (dual of [16393, 16341, 26]-code), using
- net defined by OOA [i] based on linear OOA(12852, 1366, F128, 25, 25) (dual of [(1366, 25), 34098, 26]-NRT-code), using
(27, 51, 5633)-Net over F128 — Digital
Digital (27, 51, 5633)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12851, 5633, F128, 2, 24) (dual of [(5633, 2), 11215, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12851, 8199, F128, 2, 24) (dual of [(8199, 2), 16347, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12851, 16398, F128, 24) (dual of [16398, 16347, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(12851, 16398, F128, 24) (dual of [16398, 16347, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(12851, 8199, F128, 2, 24) (dual of [(8199, 2), 16347, 25]-NRT-code), using
(27, 51, large)-Net in Base 128 — Upper bound on s
There is no (27, 51, large)-net in base 128, because
- 22 times m-reduction [i] would yield (27, 29, large)-net in base 128, but