Best Known (58−24, 58, s)-Nets in Base 128
(58−24, 58, 1368)-Net over F128 — Constructive and digital
Digital (34, 58, 1368)-net over F128, using
- 1281 times duplication [i] based on digital (33, 57, 1368)-net over F128, using
- net defined by OOA [i] based on linear OOA(12857, 1368, F128, 24, 24) (dual of [(1368, 24), 32775, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12857, 16416, F128, 24) (dual of [16416, 16359, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(12) [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(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12810, 32, F128, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(23) ⊂ Ce(12) [i] based on
- OA 12-folding and stacking [i] based on linear OA(12857, 16416, F128, 24) (dual of [16416, 16359, 25]-code), using
- net defined by OOA [i] based on linear OOA(12857, 1368, F128, 24, 24) (dual of [(1368, 24), 32775, 25]-NRT-code), using
(58−24, 58, 5462)-Net in Base 128 — Constructive
(34, 58, 5462)-net in base 128, using
- 1282 times duplication [i] based on (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
- base change [i] based on digital (25, 49, 5462)-net over F256, using
(58−24, 58, 16419)-Net over F128 — Digital
Digital (34, 58, 16419)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12858, 16419, F128, 24) (dual of [16419, 16361, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(11) [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(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12811, 35, F128, 11) (dual of [35, 24, 12]-code or 35-arc in PG(10,128)), using
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- Reed–Solomon code RS(117,128) [i]
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- construction X applied to Ce(23) ⊂ Ce(11) [i] based on
(58−24, 58, large)-Net in Base 128 — Upper bound on s
There is no (34, 58, large)-net in base 128, because
- 22 times m-reduction [i] would yield (34, 36, large)-net in base 128, but