Best Known (55−26, 55, s)-Nets in Base 128
(55−26, 55, 1261)-Net over F128 — Constructive and digital
Digital (29, 55, 1261)-net over F128, using
- 1 times m-reduction [i] based on digital (29, 56, 1261)-net over F128, using
- net defined by OOA [i] based on linear OOA(12856, 1261, F128, 27, 27) (dual of [(1261, 27), 33991, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12856, 16394, F128, 27) (dual of [16394, 16338, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12856, 16396, F128, 27) (dual of [16396, 16340, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [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(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-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([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12856, 16396, F128, 27) (dual of [16396, 16340, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12856, 16394, F128, 27) (dual of [16394, 16338, 28]-code), using
- net defined by OOA [i] based on linear OOA(12856, 1261, F128, 27, 27) (dual of [(1261, 27), 33991, 28]-NRT-code), using
(55−26, 55, 5466)-Net over F128 — Digital
Digital (29, 55, 5466)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12855, 5466, F128, 3, 26) (dual of [(5466, 3), 16343, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12855, 16398, F128, 26) (dual of [16398, 16343, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- OOA 3-folding [i] based on linear OA(12855, 16398, F128, 26) (dual of [16398, 16343, 27]-code), using
(55−26, 55, large)-Net in Base 128 — Upper bound on s
There is no (29, 55, large)-net in base 128, because
- 24 times m-reduction [i] would yield (29, 31, large)-net in base 128, but