Best Known (34, 61, s)-Nets in Base 128
(34, 61, 1262)-Net over F128 — Constructive and digital
Digital (34, 61, 1262)-net over F128, using
- 1281 times duplication [i] based on digital (33, 60, 1262)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 1262, F128, 27, 27) (dual of [(1262, 27), 34014, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12860, 16407, F128, 27) (dual of [16407, 16347, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 16408, F128, 27) (dual of [16408, 16348, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- linear OA(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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(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([0,13]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 16408, F128, 27) (dual of [16408, 16348, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12860, 16407, F128, 27) (dual of [16407, 16347, 28]-code), using
- net defined by OOA [i] based on linear OOA(12860, 1262, F128, 27, 27) (dual of [(1262, 27), 34014, 28]-NRT-code), using
(34, 61, 5041)-Net in Base 128 — Constructive
(34, 61, 5041)-net in base 128, using
- net defined by OOA [i] based on OOA(12861, 5041, S128, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(12861, 65534, S128, 27), using
- discarding factors based on OA(12861, 65538, S128, 27), using
- discarding parts of the base [i] based on linear OA(25653, 65538, F256, 27) (dual of [65538, 65485, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(25653, 65536, F256, 27) (dual of [65536, 65483, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- discarding parts of the base [i] based on linear OA(25653, 65538, F256, 27) (dual of [65538, 65485, 28]-code), using
- discarding factors based on OA(12861, 65538, S128, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(12861, 65534, S128, 27), using
(34, 61, 9132)-Net over F128 — Digital
Digital (34, 61, 9132)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12861, 9132, F128, 27) (dual of [9132, 9071, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 16410, F128, 27) (dual of [16410, 16349, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(17) [i] based on
- linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(26) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12861, 16410, F128, 27) (dual of [16410, 16349, 28]-code), using
(34, 61, large)-Net in Base 128 — Upper bound on s
There is no (34, 61, large)-net in base 128, because
- 25 times m-reduction [i] would yield (34, 36, large)-net in base 128, but