Best Known (61−13, 61, s)-Nets in Base 8
(61−13, 61, 5465)-Net over F8 — Constructive and digital
Digital (48, 61, 5465)-net over F8, using
- 81 times duplication [i] based on digital (47, 60, 5465)-net over F8, using
- net defined by OOA [i] based on linear OOA(860, 5465, F8, 13, 13) (dual of [(5465, 13), 70985, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(860, 32791, F8, 13) (dual of [32791, 32731, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(860, 32792, F8, 13) (dual of [32792, 32732, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- 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(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(84, 24, F8, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,8)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(860, 32792, F8, 13) (dual of [32792, 32732, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(860, 32791, F8, 13) (dual of [32791, 32731, 14]-code), using
- net defined by OOA [i] based on linear OOA(860, 5465, F8, 13, 13) (dual of [(5465, 13), 70985, 14]-NRT-code), using
(61−13, 61, 32794)-Net over F8 — Digital
Digital (48, 61, 32794)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(861, 32794, F8, 13) (dual of [32794, 32733, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(860, 32792, F8, 13) (dual of [32792, 32732, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- 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(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(84, 24, F8, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,8)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(860, 32793, F8, 12) (dual of [32793, 32733, 13]-code), using Gilbert–Varšamov bound and bm = 860 > Vbs−1(k−1) = 2331 642731 932600 130325 166091 423441 526316 248204 768498 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(860, 32792, F8, 13) (dual of [32792, 32732, 14]-code), using
- construction X with Varšamov bound [i] based on
(61−13, 61, large)-Net in Base 8 — Upper bound on s
There is no (48, 61, large)-net in base 8, because
- 11 times m-reduction [i] would yield (48, 50, large)-net in base 8, but