Best Known (48, 48+14, s)-Nets in Base 8
(48, 48+14, 4682)-Net over F8 — Constructive and digital
Digital (48, 62, 4682)-net over F8, using
- net defined by OOA [i] based on linear OOA(862, 4682, F8, 14, 14) (dual of [(4682, 14), 65486, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(862, 32774, F8, 14) (dual of [32774, 32712, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(862, 32779, F8, 14) (dual of [32779, 32717, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(862, 32779, F8, 14) (dual of [32779, 32717, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(862, 32774, F8, 14) (dual of [32774, 32712, 15]-code), using
(48, 48+14, 29436)-Net over F8 — Digital
Digital (48, 62, 29436)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(862, 29436, F8, 14) (dual of [29436, 29374, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(862, 32779, F8, 14) (dual of [32779, 32717, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(862, 32779, F8, 14) (dual of [32779, 32717, 15]-code), using
(48, 48+14, large)-Net in Base 8 — Upper bound on s
There is no (48, 62, large)-net in base 8, because
- 12 times m-reduction [i] would yield (48, 50, large)-net in base 8, but