Best Known (35, 35+33, s)-Nets in Base 128
(35, 35+33, 1024)-Net over F128 — Constructive and digital
Digital (35, 68, 1024)-net over F128, using
- 1283 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
(35, 35+33, 4843)-Net over F128 — Digital
Digital (35, 68, 4843)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12868, 4843, F128, 3, 33) (dual of [(4843, 3), 14461, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12868, 5465, F128, 3, 33) (dual of [(5465, 3), 16327, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12868, 16395, F128, 33) (dual of [16395, 16327, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(12868, 16396, F128, 33) (dual of [16396, 16328, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [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]
- linear OA(12857, 16385, F128, 29) (dual of [16385, 16328, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12868, 16396, F128, 33) (dual of [16396, 16328, 34]-code), using
- OOA 3-folding [i] based on linear OA(12868, 16395, F128, 33) (dual of [16395, 16327, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(12868, 5465, F128, 3, 33) (dual of [(5465, 3), 16327, 34]-NRT-code), using
(35, 35+33, large)-Net in Base 128 — Upper bound on s
There is no (35, 68, large)-net in base 128, because
- 31 times m-reduction [i] would yield (35, 37, large)-net in base 128, but