Best Known (62−13, 62, s)-Nets in Base 128
(62−13, 62, 1403564)-Net over F128 — Constructive and digital
Digital (49, 62, 1403564)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 5464)-net over F128, using
- net defined by OOA [i] based on linear OOA(12813, 5464, F128, 6, 6) (dual of [(5464, 6), 32771, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12813, 16392, F128, 6) (dual of [16392, 16379, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1285, 16384, F128, 3) (dual of [16384, 16379, 4]-code or 16384-cap in PG(4,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(12813, 16392, F128, 6) (dual of [16392, 16379, 7]-code), using
- net defined by OOA [i] based on linear OOA(12813, 5464, F128, 6, 6) (dual of [(5464, 6), 32771, 7]-NRT-code), using
- digital (36, 49, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (7, 13, 5464)-net over F128, using
(62−13, 62, 2796200)-Net in Base 128 — Constructive
(49, 62, 2796200)-net in base 128, using
- 1281 times duplication [i] based on (48, 61, 2796200)-net in base 128, using
- net defined by OOA [i] based on OOA(12861, 2796200, S128, 14, 13), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12861, 8388601, S128, 2, 13), using
- discarding factors based on OOA(12861, 8388602, S128, 2, 13), using
- discarding parts of the base [i] based on linear OOA(25653, 8388602, F256, 2, 13) (dual of [(8388602, 2), 16777151, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(25616, 4194301, F256, 2, 6) (dual of [(4194301, 2), 8388586, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25616, 8388602, F256, 6) (dual of [8388602, 8388586, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- OOA 2-folding [i] based on linear OA(25616, 8388602, F256, 6) (dual of [8388602, 8388586, 7]-code), using
- linear OOA(25637, 4194301, F256, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- linear OOA(25616, 4194301, F256, 2, 6) (dual of [(4194301, 2), 8388586, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25653, 8388602, F256, 2, 13) (dual of [(8388602, 2), 16777151, 14]-NRT-code), using
- discarding factors based on OOA(12861, 8388602, S128, 2, 13), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12861, 8388601, S128, 2, 13), using
- net defined by OOA [i] based on OOA(12861, 2796200, S128, 14, 13), using
(62−13, 62, large)-Net over F128 — Digital
Digital (49, 62, large)-net over F128, using
- t-expansion [i] based on digital (48, 62, large)-net over F128, using
- 4 times m-reduction [i] based on digital (48, 66, large)-net over F128, using
(62−13, 62, large)-Net in Base 128 — Upper bound on s
There is no (49, 62, large)-net in base 128, because
- 11 times m-reduction [i] would yield (49, 51, large)-net in base 128, but