Best Known (67, 67+13, s)-Nets in Base 8
(67, 67+13, 349528)-Net over F8 — Constructive and digital
Digital (67, 80, 349528)-net over F8, using
- net defined by OOA [i] based on linear OOA(880, 349528, F8, 13, 13) (dual of [(349528, 13), 4543784, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(880, 2097169, F8, 13) (dual of [2097169, 2097089, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(879, 2097168, F8, 13) (dual of [2097168, 2097089, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(815, 16, F8, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
- dual of repetition code with length 16 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(879, 2097168, F8, 13) (dual of [2097168, 2097089, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(880, 2097169, F8, 13) (dual of [2097169, 2097089, 14]-code), using
(67, 67+13, 2097170)-Net over F8 — Digital
Digital (67, 80, 2097170)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(880, 2097170, F8, 13) (dual of [2097170, 2097090, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(879, 2097168, F8, 13) (dual of [2097168, 2097089, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(878, 2097152, F8, 13) (dual of [2097152, 2097074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(815, 16, F8, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
- dual of repetition code with length 16 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(879, 2097169, F8, 12) (dual of [2097169, 2097090, 13]-code), using Gilbert–Varšamov bound and bm = 879 > Vbs−1(k−1) = 170953 148390 081251 861356 815602 110868 534157 382605 927685 481345 118618 666283 [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(879, 2097168, F8, 13) (dual of [2097168, 2097089, 14]-code), using
- construction X with Varšamov bound [i] based on
(67, 67+13, large)-Net in Base 8 — Upper bound on s
There is no (67, 80, large)-net in base 8, because
- 11 times m-reduction [i] would yield (67, 69, large)-net in base 8, but