Best Known (26−10, 26, s)-Nets in Base 128
(26−10, 26, 6554)-Net over F128 — Constructive and digital
Digital (16, 26, 6554)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- appending kth column [i] based on linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
- digital (9, 19, 3277)-net over F128, using
- net defined by OOA [i] based on linear OOA(12819, 3277, F128, 10, 10) (dual of [(3277, 10), 32751, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12819, 16385, F128, 10) (dual of [16385, 16366, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(12819, 16385, F128, 10) (dual of [16385, 16366, 11]-code), using
- net defined by OOA [i] based on linear OOA(12819, 3277, F128, 10, 10) (dual of [(3277, 10), 32751, 11]-NRT-code), using
- digital (2, 7, 8128)-net over F128, using
(26−10, 26, 13109)-Net in Base 128 — Constructive
(16, 26, 13109)-net in base 128, using
- 1281 times duplication [i] based on (15, 25, 13109)-net in base 128, using
- net defined by OOA [i] based on OOA(12825, 13109, S128, 10, 10), using
- OA 5-folding and stacking [i] based on OA(12825, 65545, S128, 10), using
- 1 times code embedding in larger space [i] based on OA(12824, 65544, S128, 10), using
- discarding parts of the base [i] based on linear OA(25621, 65544, F256, 10) (dual of [65544, 65523, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(25621, 65544, F256, 10) (dual of [65544, 65523, 11]-code), using
- 1 times code embedding in larger space [i] based on OA(12824, 65544, S128, 10), using
- OA 5-folding and stacking [i] based on OA(12825, 65545, S128, 10), using
- net defined by OOA [i] based on OOA(12825, 13109, S128, 10, 10), using
(26−10, 26, 39948)-Net over F128 — Digital
Digital (16, 26, 39948)-net over F128, using
(26−10, 26, large)-Net in Base 128 — Upper bound on s
There is no (16, 26, large)-net in base 128, because
- 8 times m-reduction [i] would yield (16, 18, large)-net in base 128, but