Best Known (31−8, 31, s)-Nets in Base 8
(31−8, 31, 2049)-Net over F8 — Constructive and digital
Digital (23, 31, 2049)-net over F8, using
- 81 times duplication [i] based on digital (22, 30, 2049)-net over F8, using
- net defined by OOA [i] based on linear OOA(830, 2049, F8, 8, 8) (dual of [(2049, 8), 16362, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(830, 8196, F8, 8) (dual of [8196, 8166, 9]-code), using
- trace code [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(830, 8196, F8, 8) (dual of [8196, 8166, 9]-code), using
- net defined by OOA [i] based on linear OOA(830, 2049, F8, 8, 8) (dual of [(2049, 8), 16362, 9]-NRT-code), using
(31−8, 31, 8198)-Net over F8 — Digital
Digital (23, 31, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(831, 8198, F8, 8) (dual of [8198, 8167, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(830, 8196, F8, 8) (dual of [8196, 8166, 9]-code), using
- trace code [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- linear OA(830, 8197, F8, 7) (dual of [8197, 8167, 8]-code), using Gilbert–Varšamov bound and bm = 830 > Vbs−1(k−1) = 49 444487 922665 468297 316352 [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(830, 8196, F8, 8) (dual of [8196, 8166, 9]-code), using
- construction X with Varšamov bound [i] based on
(31−8, 31, 3154291)-Net in Base 8 — Upper bound on s
There is no (23, 31, 3154292)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 9903 521234 134113 382349 486172 > 831 [i]