Best Known (121−27, 121, s)-Nets in Base 8
(121−27, 121, 2522)-Net over F8 — Constructive and digital
Digital (94, 121, 2522)-net over F8, using
- 81 times duplication [i] based on digital (93, 120, 2522)-net over F8, using
- net defined by OOA [i] based on linear OOA(8120, 2522, F8, 27, 27) (dual of [(2522, 27), 67974, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8120, 32787, F8, 27) (dual of [32787, 32667, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(8116, 32768, F8, 27) (dual of [32768, 32652, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8101, 32768, F8, 23) (dual of [32768, 32667, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(26) ⊂ Ce(22) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(8120, 32787, F8, 27) (dual of [32787, 32667, 28]-code), using
- net defined by OOA [i] based on linear OOA(8120, 2522, F8, 27, 27) (dual of [(2522, 27), 67974, 28]-NRT-code), using
(121−27, 121, 31416)-Net over F8 — Digital
Digital (94, 121, 31416)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8121, 31416, F8, 27) (dual of [31416, 31295, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8121, 32769, F8, 27) (dual of [32769, 32648, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8121, 32769, F8, 27) (dual of [32769, 32648, 28]-code), using
(121−27, 121, large)-Net in Base 8 — Upper bound on s
There is no (94, 121, large)-net in base 8, because
- 25 times m-reduction [i] would yield (94, 96, large)-net in base 8, but