Best Known (35, 52, s)-Nets in Base 128
(35, 52, 262145)-Net over F128 — Constructive and digital
Digital (35, 52, 262145)-net over F128, using
- 1281 times duplication [i] based on digital (34, 51, 262145)-net over F128, using
- net defined by OOA [i] based on linear OOA(12851, 262145, F128, 17, 17) (dual of [(262145, 17), 4456414, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12851, 2097161, F128, 17) (dual of [2097161, 2097110, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12851, 2097163, F128, 17) (dual of [2097163, 2097112, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(12851, 2097163, F128, 17) (dual of [2097163, 2097112, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12851, 2097161, F128, 17) (dual of [2097161, 2097110, 18]-code), using
- net defined by OOA [i] based on linear OOA(12851, 262145, F128, 17, 17) (dual of [(262145, 17), 4456414, 18]-NRT-code), using
(35, 52, 1048584)-Net over F128 — Digital
Digital (35, 52, 1048584)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12852, 1048584, F128, 2, 17) (dual of [(1048584, 2), 2097116, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12852, 2097168, F128, 17) (dual of [2097168, 2097116, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-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([0,8]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(12852, 2097168, F128, 17) (dual of [2097168, 2097116, 18]-code), using
(35, 52, large)-Net in Base 128 — Upper bound on s
There is no (35, 52, large)-net in base 128, because
- 15 times m-reduction [i] would yield (35, 37, large)-net in base 128, but