Best Known (77−35, 77, s)-Nets in Base 128
(77−35, 77, 965)-Net over F128 — Constructive and digital
Digital (42, 77, 965)-net over F128, using
- 1281 times duplication [i] based on digital (41, 76, 965)-net over F128, using
- net defined by OOA [i] based on linear OOA(12876, 965, F128, 35, 35) (dual of [(965, 35), 33699, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12876, 16406, F128, 35) (dual of [16406, 16330, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12876, 16408, F128, 35) (dual of [16408, 16332, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- linear OA(12869, 16385, F128, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- 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(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([0,17]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12876, 16408, F128, 35) (dual of [16408, 16332, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12876, 16406, F128, 35) (dual of [16406, 16330, 36]-code), using
- net defined by OOA [i] based on linear OOA(12876, 965, F128, 35, 35) (dual of [(965, 35), 33699, 36]-NRT-code), using
(77−35, 77, 8205)-Net over F128 — Digital
Digital (42, 77, 8205)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12877, 8205, F128, 2, 35) (dual of [(8205, 2), 16333, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12877, 16410, F128, 35) (dual of [16410, 16333, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(25) [i] based on
- linear OA(12869, 16384, F128, 35) (dual of [16384, 16315, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- 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(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(34) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(12877, 16410, F128, 35) (dual of [16410, 16333, 36]-code), using
(77−35, 77, large)-Net in Base 128 — Upper bound on s
There is no (42, 77, large)-net in base 128, because
- 33 times m-reduction [i] would yield (42, 44, large)-net in base 128, but