Best Known (37, 37+29, s)-Nets in Base 128
(37, 37+29, 1172)-Net over F128 — Constructive and digital
Digital (37, 66, 1172)-net over F128, using
- 1281 times duplication [i] based on digital (36, 65, 1172)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 1172, F128, 29, 29) (dual of [(1172, 29), 33923, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12865, 16409, F128, 29) (dual of [16409, 16344, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(12865, 16410, F128, 29) (dual of [16410, 16345, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(19) [i] based on
- linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1288, 26, F128, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(28) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(12865, 16410, F128, 29) (dual of [16410, 16345, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12865, 16409, F128, 29) (dual of [16409, 16344, 30]-code), using
- net defined by OOA [i] based on linear OOA(12865, 1172, F128, 29, 29) (dual of [(1172, 29), 33923, 30]-NRT-code), using
(37, 37+29, 4681)-Net in Base 128 — Constructive
(37, 66, 4681)-net in base 128, using
- net defined by OOA [i] based on OOA(12866, 4681, S128, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12866, 65535, S128, 29), using
- discarding factors based on OA(12866, 65538, S128, 29), using
- discarding parts of the base [i] based on linear OA(25657, 65538, F256, 29) (dual of [65538, 65481, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- discarding parts of the base [i] based on linear OA(25657, 65538, F256, 29) (dual of [65538, 65481, 30]-code), using
- discarding factors based on OA(12866, 65538, S128, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12866, 65535, S128, 29), using
(37, 37+29, 10162)-Net over F128 — Digital
Digital (37, 66, 10162)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12866, 10162, F128, 29) (dual of [10162, 10096, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(12866, 16414, F128, 29) (dual of [16414, 16348, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,9]) [i] based on
- 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(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,14]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12866, 16414, F128, 29) (dual of [16414, 16348, 30]-code), using
(37, 37+29, large)-Net in Base 128 — Upper bound on s
There is no (37, 66, large)-net in base 128, because
- 27 times m-reduction [i] would yield (37, 39, large)-net in base 128, but