Best Known (39, 39+21, s)-Nets in Base 128
(39, 39+21, 3277)-Net over F128 — Constructive and digital
Digital (39, 60, 3277)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 3277, F128, 21, 21) (dual of [(3277, 21), 68757, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12860, 32771, F128, 21) (dual of [32771, 32711, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 32772, F128, 21) (dual of [32772, 32712, 22]-code), using
- (u, u+v)-construction [i] 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
- linear OA(12841, 16386, F128, 21) (dual of [16386, 16345, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code) (see above)
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 32772, F128, 21) (dual of [32772, 32712, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12860, 32771, F128, 21) (dual of [32771, 32711, 22]-code), using
(39, 39+21, 6811)-Net in Base 128 — Constructive
(39, 60, 6811)-net in base 128, using
- (u, u+v)-construction [i] based on
- (3, 13, 258)-net in base 128, using
- 3 times m-reduction [i] based on (3, 16, 258)-net in base 128, using
- base change [i] based on digital (1, 14, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 14, 258)-net over F256, using
- 3 times m-reduction [i] based on (3, 16, 258)-net in base 128, using
- (26, 47, 6553)-net in base 128, using
- net defined by OOA [i] based on OOA(12847, 6553, S128, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12847, 65531, S128, 21), using
- discarding factors based on OA(12847, 65538, S128, 21), using
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- discarding factors based on OA(12847, 65538, S128, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12847, 65531, S128, 21), using
- net defined by OOA [i] based on OOA(12847, 6553, S128, 21, 21), using
- (3, 13, 258)-net in base 128, using
(39, 39+21, 137140)-Net over F128 — Digital
Digital (39, 60, 137140)-net over F128, using
(39, 39+21, large)-Net in Base 128 — Upper bound on s
There is no (39, 60, large)-net in base 128, because
- 19 times m-reduction [i] would yield (39, 41, large)-net in base 128, but