Best Known (41, 41+29, s)-Nets in Base 128
(41, 41+29, 1173)-Net over F128 — Constructive and digital
Digital (41, 70, 1173)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 1173, F128, 29, 29) (dual of [(1173, 29), 33947, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12870, 16423, F128, 29) (dual of [16423, 16353, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16426, F128, 29) (dual of [16426, 16356, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,7]) [i] based on
- linear OA(12857, 16385, F128, 29) (dual of [16385, 16328, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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(12813, 41, F128, 13) (dual of [41, 28, 14]-code or 41-arc in PG(12,128)), using
- discarding factors / shortening the dual code based on linear OA(12813, 128, F128, 13) (dual of [128, 115, 14]-code or 128-arc in PG(12,128)), using
- Reed–Solomon code RS(115,128) [i]
- discarding factors / shortening the dual code based on linear OA(12813, 128, F128, 13) (dual of [128, 115, 14]-code or 128-arc in PG(12,128)), using
- construction X applied to C([0,14]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16426, F128, 29) (dual of [16426, 16356, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12870, 16423, F128, 29) (dual of [16423, 16353, 30]-code), using
(41, 41+29, 4682)-Net in Base 128 — Constructive
(41, 70, 4682)-net in base 128, using
- net defined by OOA [i] based on OOA(12870, 4682, S128, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12870, 65549, S128, 29), using
- discarding factors based on OA(12870, 65550, S128, 29), using
- discarding parts of the base [i] based on linear OA(25661, 65550, F256, 29) (dual of [65550, 65489, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- discarding parts of the base [i] based on linear OA(25661, 65550, F256, 29) (dual of [65550, 65489, 30]-code), using
- discarding factors based on OA(12870, 65550, S128, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12870, 65549, S128, 29), using
(41, 41+29, 16504)-Net over F128 — Digital
Digital (41, 70, 16504)-net over F128, using
(41, 41+29, large)-Net in Base 128 — Upper bound on s
There is no (41, 70, large)-net in base 128, because
- 27 times m-reduction [i] would yield (41, 43, large)-net in base 128, but