Best Known (48−23, 48, s)-Nets in Base 128
(48−23, 48, 1490)-Net over F128 — Constructive and digital
Digital (25, 48, 1490)-net over F128, using
- 1281 times duplication [i] based on digital (24, 47, 1490)-net over F128, using
- net defined by OOA [i] based on linear OOA(12847, 1490, F128, 23, 23) (dual of [(1490, 23), 34223, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12847, 16391, F128, 23) (dual of [16391, 16344, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12847, 16392, F128, 23) (dual of [16392, 16345, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(12847, 16392, F128, 23) (dual of [16392, 16345, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12847, 16391, F128, 23) (dual of [16391, 16344, 24]-code), using
- net defined by OOA [i] based on linear OOA(12847, 1490, F128, 23, 23) (dual of [(1490, 23), 34223, 24]-NRT-code), using
(48−23, 48, 5465)-Net over F128 — Digital
Digital (25, 48, 5465)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12848, 5465, F128, 3, 23) (dual of [(5465, 3), 16347, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12848, 16395, F128, 23) (dual of [16395, 16347, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 16396, F128, 23) (dual of [16396, 16348, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- 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(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(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([0,11]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 16396, F128, 23) (dual of [16396, 16348, 24]-code), using
- OOA 3-folding [i] based on linear OA(12848, 16395, F128, 23) (dual of [16395, 16347, 24]-code), using
(48−23, 48, large)-Net in Base 128 — Upper bound on s
There is no (25, 48, large)-net in base 128, because
- 21 times m-reduction [i] would yield (25, 27, large)-net in base 128, but