Best Known (56−24, 56, s)-Nets in Base 128
(56−24, 56, 1367)-Net over F128 — Constructive and digital
Digital (32, 56, 1367)-net over F128, using
- t-expansion [i] based on digital (31, 56, 1367)-net over F128, using
- net defined by OOA [i] based on linear OOA(12856, 1367, F128, 25, 25) (dual of [(1367, 25), 34119, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12856, 16405, F128, 25) (dual of [16405, 16349, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12856, 16408, F128, 25) (dual of [16408, 16352, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12856, 16408, F128, 25) (dual of [16408, 16352, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12856, 16405, F128, 25) (dual of [16405, 16349, 26]-code), using
- net defined by OOA [i] based on linear OOA(12856, 1367, F128, 25, 25) (dual of [(1367, 25), 34119, 26]-NRT-code), using
(56−24, 56, 5462)-Net in Base 128 — Constructive
(32, 56, 5462)-net in base 128, using
- base change [i] based on digital (25, 49, 5462)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5462, F256, 24, 24) (dual of [(5462, 24), 131039, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- OA 12-folding and stacking [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5462, F256, 24, 24) (dual of [(5462, 24), 131039, 25]-NRT-code), using
(56−24, 56, 13205)-Net over F128 — Digital
Digital (32, 56, 13205)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12856, 13205, F128, 24) (dual of [13205, 13149, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12856, 16413, F128, 24) (dual of [16413, 16357, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(13) [i] based on
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1289, 29, F128, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,128)), using
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- Reed–Solomon code RS(119,128) [i]
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- construction X applied to Ce(23) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(12856, 16413, F128, 24) (dual of [16413, 16357, 25]-code), using
(56−24, 56, 15710)-Net in Base 128
(32, 56, 15710)-net in base 128, using
- base change [i] based on digital (25, 49, 15710)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 15710, F256, 4, 24) (dual of [(15710, 4), 62791, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 16386, F256, 4, 24) (dual of [(16386, 4), 65495, 25]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- OOA 4-folding [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 16386, F256, 4, 24) (dual of [(16386, 4), 65495, 25]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 15710, F256, 4, 24) (dual of [(15710, 4), 62791, 25]-NRT-code), using
(56−24, 56, large)-Net in Base 128 — Upper bound on s
There is no (32, 56, large)-net in base 128, because
- 22 times m-reduction [i] would yield (32, 34, large)-net in base 128, but