Best Known (21−10, 21, s)-Nets in Base 128
(21−10, 21, 3278)-Net over F128 — Constructive and digital
Digital (11, 21, 3278)-net over F128, using
- net defined by OOA [i] based on linear OOA(12821, 3278, F128, 10, 10) (dual of [(3278, 10), 32759, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12821, 16390, F128, 10) (dual of [16390, 16369, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12821, 16392, F128, 10) (dual of [16392, 16371, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [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(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(12821, 16392, F128, 10) (dual of [16392, 16371, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(12821, 16390, F128, 10) (dual of [16390, 16369, 11]-code), using
(21−10, 21, 8196)-Net over F128 — Digital
Digital (11, 21, 8196)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12821, 8196, F128, 2, 10) (dual of [(8196, 2), 16371, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12821, 16392, F128, 10) (dual of [16392, 16371, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [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(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(12821, 16392, F128, 10) (dual of [16392, 16371, 11]-code), using
(21−10, 21, large)-Net in Base 128 — Upper bound on s
There is no (11, 21, large)-net in base 128, because
- 8 times m-reduction [i] would yield (11, 13, large)-net in base 128, but