Best Known (65−33, 65, s)-Nets in Base 128
(65−33, 65, 1024)-Net over F128 — Constructive and digital
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
(65−33, 65, 3453)-Net over F128 — Digital
Digital (32, 65, 3453)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12865, 3453, F128, 4, 33) (dual of [(3453, 4), 13747, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12865, 4096, F128, 4, 33) (dual of [(4096, 4), 16319, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12865, 16384, F128, 33) (dual of [16384, 16319, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- OOA 4-folding [i] based on linear OA(12865, 16384, F128, 33) (dual of [16384, 16319, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(12865, 4096, F128, 4, 33) (dual of [(4096, 4), 16319, 34]-NRT-code), using
(65−33, 65, large)-Net in Base 128 — Upper bound on s
There is no (32, 65, large)-net in base 128, because
- 31 times m-reduction [i] would yield (32, 34, large)-net in base 128, but