Best Known (52, 52+15, s)-Nets in Base 8
(52, 52+15, 4682)-Net over F8 — Constructive and digital
Digital (52, 67, 4682)-net over F8, using
- net defined by OOA [i] based on linear OOA(867, 4682, F8, 15, 15) (dual of [(4682, 15), 70163, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(867, 32775, F8, 15) (dual of [32775, 32708, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(867, 32779, F8, 15) (dual of [32779, 32712, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(856, 32768, F8, 13) (dual of [32768, 32712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(867, 32779, F8, 15) (dual of [32779, 32712, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(867, 32775, F8, 15) (dual of [32775, 32708, 16]-code), using
(52, 52+15, 31126)-Net over F8 — Digital
Digital (52, 67, 31126)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(867, 31126, F8, 15) (dual of [31126, 31059, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(867, 32779, F8, 15) (dual of [32779, 32712, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(856, 32768, F8, 13) (dual of [32768, 32712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(867, 32779, F8, 15) (dual of [32779, 32712, 16]-code), using
(52, 52+15, large)-Net in Base 8 — Upper bound on s
There is no (52, 67, large)-net in base 8, because
- 13 times m-reduction [i] would yield (52, 54, large)-net in base 8, but