Best Known (45, 45+21, s)-Nets in Base 128
(45, 45+21, 209717)-Net over F128 — Constructive and digital
Digital (45, 66, 209717)-net over F128, using
- 1281 times duplication [i] based on digital (44, 65, 209717)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 209717, F128, 21, 21) (dual of [(209717, 21), 4403992, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [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(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-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(20) ⊂ Ce(15) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- net defined by OOA [i] based on linear OOA(12865, 209717, F128, 21, 21) (dual of [(209717, 21), 4403992, 22]-NRT-code), using
(45, 45+21, 1048588)-Net over F128 — Digital
Digital (45, 66, 1048588)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12866, 1048588, F128, 2, 21) (dual of [(1048588, 2), 2097110, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12866, 2097176, F128, 21) (dual of [2097176, 2097110, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [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(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- OOA 2-folding [i] based on linear OA(12866, 2097176, F128, 21) (dual of [2097176, 2097110, 22]-code), using
(45, 45+21, large)-Net in Base 128 — Upper bound on s
There is no (45, 66, large)-net in base 128, because
- 19 times m-reduction [i] would yield (45, 47, large)-net in base 128, but