Best Known (26, 26+7, s)-Nets in Base 8
(26, 26+7, 10927)-Net over F8 — Constructive and digital
Digital (26, 33, 10927)-net over F8, using
- net defined by OOA [i] based on linear OOA(833, 10927, F8, 7, 7) (dual of [(10927, 7), 76456, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(833, 32782, F8, 7) (dual of [32782, 32749, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(832, 32781, F8, 7) (dual of [32781, 32749, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(831, 32769, F8, 7) (dual of [32769, 32738, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(821, 32769, F8, 5) (dual of [32769, 32748, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(811, 12, F8, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,8)), using
- dual of repetition code with length 12 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 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 C([0,3]) ⊂ C([0,2]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(832, 32781, F8, 7) (dual of [32781, 32749, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(833, 32782, F8, 7) (dual of [32782, 32749, 8]-code), using
(26, 26+7, 39642)-Net over F8 — Digital
Digital (26, 33, 39642)-net over F8, using
(26, 26+7, large)-Net in Base 8 — Upper bound on s
There is no (26, 33, large)-net in base 8, because
- 5 times m-reduction [i] would yield (26, 28, large)-net in base 8, but