Best Known (47−21, 47, s)-Nets in Base 128
(47−21, 47, 1640)-Net over F128 — Constructive and digital
Digital (26, 47, 1640)-net over F128, using
- 1281 times duplication [i] based on digital (25, 46, 1640)-net over F128, using
- net defined by OOA [i] based on linear OOA(12846, 1640, F128, 21, 21) (dual of [(1640, 21), 34394, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12846, 16401, F128, 21) (dual of [16401, 16355, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12846, 16402, F128, 21) (dual of [16402, 16356, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-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
- discarding factors / shortening the dual code based on linear OA(12846, 16402, F128, 21) (dual of [16402, 16356, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12846, 16401, F128, 21) (dual of [16401, 16355, 22]-code), using
- net defined by OOA [i] based on linear OOA(12846, 1640, F128, 21, 21) (dual of [(1640, 21), 34394, 22]-NRT-code), using
(47−21, 47, 6553)-Net in Base 128 — Constructive
(26, 47, 6553)-net in base 128, using
- net defined by OOA [i] based on OOA(12847, 6553, S128, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12847, 65531, S128, 21), using
- discarding factors based on OA(12847, 65538, S128, 21), using
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- discarding factors based on OA(12847, 65538, S128, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12847, 65531, S128, 21), using
(47−21, 47, 8202)-Net over F128 — Digital
Digital (26, 47, 8202)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12847, 8202, F128, 2, 21) (dual of [(8202, 2), 16357, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12847, 16404, F128, 21) (dual of [16404, 16357, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- 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(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(12847, 16404, F128, 21) (dual of [16404, 16357, 22]-code), using
(47−21, 47, large)-Net in Base 128 — Upper bound on s
There is no (26, 47, large)-net in base 128, because
- 19 times m-reduction [i] would yield (26, 28, large)-net in base 128, but