Best Known (69−33, 69, s)-Nets in Base 128
(69−33, 69, 1024)-Net over F128 — Constructive and digital
Digital (36, 69, 1024)-net over F128, using
- 1284 times duplication [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
(69−33, 69, 5466)-Net over F128 — Digital
Digital (36, 69, 5466)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12869, 5466, F128, 3, 33) (dual of [(5466, 3), 16329, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12869, 16398, F128, 33) (dual of [16398, 16329, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(27) [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]
- 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(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(32) ⊂ Ce(27) [i] based on
- OOA 3-folding [i] based on linear OA(12869, 16398, F128, 33) (dual of [16398, 16329, 34]-code), using
(69−33, 69, large)-Net in Base 128 — Upper bound on s
There is no (36, 69, large)-net in base 128, because
- 31 times m-reduction [i] would yield (36, 38, large)-net in base 128, but