Best Known (65−32, 65, s)-Nets in Base 128
(65−32, 65, 1024)-Net over F128 — Constructive and digital
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
(65−32, 65, 4109)-Net over F128 — Digital
Digital (33, 65, 4109)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12865, 4109, F128, 3, 32) (dual of [(4109, 3), 12262, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12865, 5464, F128, 3, 32) (dual of [(5464, 3), 16327, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12865, 16392, F128, 32) (dual of [16392, 16327, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [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(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- OOA 3-folding [i] based on linear OA(12865, 16392, F128, 32) (dual of [16392, 16327, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(12865, 5464, F128, 3, 32) (dual of [(5464, 3), 16327, 33]-NRT-code), using
(65−32, 65, large)-Net in Base 128 — Upper bound on s
There is no (33, 65, large)-net in base 128, because
- 30 times m-reduction [i] would yield (33, 35, large)-net in base 128, but