Best Known (63, 81, s)-Nets in Base 8
(63, 81, 3643)-Net over F8 — Constructive and digital
Digital (63, 81, 3643)-net over F8, using
- 81 times duplication [i] based on digital (62, 80, 3643)-net over F8, using
- net defined by OOA [i] based on linear OOA(880, 3643, F8, 18, 18) (dual of [(3643, 18), 65494, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(880, 32787, F8, 18) (dual of [32787, 32707, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(84, 19, F8, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,8)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- OA 9-folding and stacking [i] based on linear OA(880, 32787, F8, 18) (dual of [32787, 32707, 19]-code), using
- net defined by OOA [i] based on linear OOA(880, 3643, F8, 18, 18) (dual of [(3643, 18), 65494, 19]-NRT-code), using
(63, 81, 31825)-Net over F8 — Digital
Digital (63, 81, 31825)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(881, 31825, F8, 18) (dual of [31825, 31744, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(881, 32789, F8, 18) (dual of [32789, 32708, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(84, 20, F8, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,8)), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(881, 32789, F8, 18) (dual of [32789, 32708, 19]-code), using
(63, 81, large)-Net in Base 8 — Upper bound on s
There is no (63, 81, large)-net in base 8, because
- 16 times m-reduction [i] would yield (63, 65, large)-net in base 8, but