Best Known (9, 9+8, s)-Nets in Base 128
(9, 9+8, 4098)-Net over F128 — Constructive and digital
Digital (9, 17, 4098)-net over F128, using
- net defined by OOA [i] based on linear OOA(12817, 4098, F128, 8, 8) (dual of [(4098, 8), 32767, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(12817, 16392, F128, 8) (dual of [16392, 16375, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1289, 16384, F128, 5) (dual of [16384, 16375, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- OA 4-folding and stacking [i] based on linear OA(12817, 16392, F128, 8) (dual of [16392, 16375, 9]-code), using
(9, 9+8, 9807)-Net over F128 — Digital
Digital (9, 17, 9807)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12817, 9807, F128, 8) (dual of [9807, 9790, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(12817, 16392, F128, 8) (dual of [16392, 16375, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1289, 16384, F128, 5) (dual of [16384, 16375, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(12817, 16392, F128, 8) (dual of [16392, 16375, 9]-code), using
(9, 9+8, large)-Net in Base 128 — Upper bound on s
There is no (9, 17, large)-net in base 128, because
- 6 times m-reduction [i] would yield (9, 11, large)-net in base 128, but