Best Known (34, 66, s)-Nets in Base 128
(34, 66, 1024)-Net over F128 — Constructive and digital
Digital (34, 66, 1024)-net over F128, using
- 1281 times duplication [i] based on digital (33, 65, 1024)-net over F128, using
- t-expansion [i] based on digital (32, 65, 1024)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 1024, F128, 33, 33) (dual of [(1024, 33), 33727, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using
- net defined by OOA [i] based on linear OOA(12865, 1024, F128, 33, 33) (dual of [(1024, 33), 33727, 34]-NRT-code), using
- t-expansion [i] based on digital (32, 65, 1024)-net over F128, using
(34, 66, 4889)-Net over F128 — Digital
Digital (34, 66, 4889)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12866, 4889, F128, 3, 32) (dual of [(4889, 3), 14601, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12866, 5465, F128, 3, 32) (dual of [(5465, 3), 16329, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12866, 16395, F128, 32) (dual of [16395, 16329, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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 Ce(31) ⊂ Ce(27) [i] based on
- OOA 3-folding [i] based on linear OA(12866, 16395, F128, 32) (dual of [16395, 16329, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(12866, 5465, F128, 3, 32) (dual of [(5465, 3), 16329, 33]-NRT-code), using
(34, 66, large)-Net in Base 128 — Upper bound on s
There is no (34, 66, large)-net in base 128, because
- 30 times m-reduction [i] would yield (34, 36, large)-net in base 128, but