Best Known (69−21, 69, s)-Nets in Base 128
(69−21, 69, 209718)-Net over F128 — Constructive and digital
Digital (48, 69, 209718)-net over F128, using
- 1281 times duplication [i] based on digital (47, 68, 209718)-net over F128, using
- net defined by OOA [i] based on linear OOA(12868, 209718, F128, 21, 21) (dual of [(209718, 21), 4404010, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12868, 2097181, F128, 21) (dual of [2097181, 2097113, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12868, 2097184, F128, 21) (dual of [2097184, 2097116, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-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([0,10]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12868, 2097184, F128, 21) (dual of [2097184, 2097116, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12868, 2097181, F128, 21) (dual of [2097181, 2097113, 22]-code), using
- net defined by OOA [i] based on linear OOA(12868, 209718, F128, 21, 21) (dual of [(209718, 21), 4404010, 22]-NRT-code), using
(69−21, 69, 2097187)-Net over F128 — Digital
Digital (48, 69, 2097187)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12869, 2097187, F128, 21) (dual of [2097187, 2097118, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(11) [i] based on
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1288, 35, F128, 8) (dual of [35, 27, 9]-code or 35-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(20) ⊂ Ce(11) [i] based on
(69−21, 69, large)-Net in Base 128 — Upper bound on s
There is no (48, 69, large)-net in base 128, because
- 19 times m-reduction [i] would yield (48, 50, large)-net in base 128, but