Best Known (27, 27+10, s)-Nets in Base 8
(27, 27+10, 822)-Net over F8 — Constructive and digital
Digital (27, 37, 822)-net over F8, using
- net defined by OOA [i] based on linear OOA(837, 822, F8, 10, 10) (dual of [(822, 10), 8183, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(837, 4110, F8, 10) (dual of [4110, 4073, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(837, 4112, F8, 10) (dual of [4112, 4075, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(84, 16, F8, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,8)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(837, 4112, F8, 10) (dual of [4112, 4075, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(837, 4110, F8, 10) (dual of [4110, 4073, 11]-code), using
(27, 27+10, 4252)-Net over F8 — Digital
Digital (27, 37, 4252)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(837, 4252, F8, 10) (dual of [4252, 4215, 11]-code), using
- 148 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 32 times 0, 1, 107 times 0) [i] based on linear OA(833, 4100, F8, 10) (dual of [4100, 4067, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 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(80, 4, F8, 0) (dual of [4, 4, 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
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- 148 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 32 times 0, 1, 107 times 0) [i] based on linear OA(833, 4100, F8, 10) (dual of [4100, 4067, 11]-code), using
(27, 27+10, 1793095)-Net in Base 8 — Upper bound on s
There is no (27, 37, 1793096)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 2596 151006 116726 876286 411374 853051 > 837 [i]