Best Known (23, 23+9, s)-Nets in Base 8
(23, 23+9, 1026)-Net over F8 — Constructive and digital
Digital (23, 32, 1026)-net over F8, using
- 81 times duplication [i] based on digital (22, 31, 1026)-net over F8, using
- net defined by OOA [i] based on linear OOA(831, 1026, F8, 9, 9) (dual of [(1026, 9), 9203, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(831, 4105, F8, 9) (dual of [4105, 4074, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(821, 4096, F8, 6) (dual of [4096, 4075, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(82, 9, F8, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
- extended Reed–Solomon code RSe(7,8) [i]
- Hamming code H(2,8) [i]
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(831, 4105, F8, 9) (dual of [4105, 4074, 10]-code), using
- net defined by OOA [i] based on linear OOA(831, 1026, F8, 9, 9) (dual of [(1026, 9), 9203, 10]-NRT-code), using
(23, 23+9, 4139)-Net over F8 — Digital
Digital (23, 32, 4139)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(832, 4139, F8, 9) (dual of [4139, 4107, 10]-code), using
- 40 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 32 times 0) [i] based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 40 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 32 times 0) [i] based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
(23, 23+9, 3154291)-Net in Base 8 — Upper bound on s
There is no (23, 32, 3154292)-net in base 8, because
- 1 times m-reduction [i] would yield (23, 31, 3154292)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 9903 521234 134113 382349 486172 > 831 [i]