Best Known (10, 10+4, s)-Nets in Base 8
(10, 10+4, 4098)-Net over F8 — Constructive and digital
Digital (10, 14, 4098)-net over F8, using
- net defined by OOA [i] based on linear OOA(814, 4098, F8, 4, 4) (dual of [(4098, 4), 16378, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(814, 4098, F8, 3, 4) (dual of [(4098, 3), 12280, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(814, 8196, F8, 4) (dual of [8196, 8182, 5]-code), using
- trace code [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(814, 8196, F8, 4) (dual of [8196, 8182, 5]-code), using
- appending kth column [i] based on linear OOA(814, 4098, F8, 3, 4) (dual of [(4098, 3), 12280, 5]-NRT-code), using
(10, 10+4, 8196)-Net over F8 — Digital
Digital (10, 14, 8196)-net over F8, using
- net defined by OOA [i] based on linear OOA(814, 8196, F8, 4, 4) (dual of [(8196, 4), 32770, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(814, 8196, F8, 3, 4) (dual of [(8196, 3), 24574, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(814, 8196, F8, 4) (dual of [8196, 8182, 5]-code), using
- trace code [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(814, 8196, F8, 4) (dual of [8196, 8182, 5]-code), using
- appending kth column [i] based on linear OOA(814, 8196, F8, 3, 4) (dual of [(8196, 3), 24574, 5]-NRT-code), using
(10, 10+4, 16256)-Net in Base 8 — Constructive
(10, 14, 16256)-net in base 8, using
- trace code for nets [i] based on (3, 7, 8128)-net in base 64, using
- base change [i] based on digital (2, 6, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- 1 times truncation [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- base change [i] based on digital (2, 6, 8128)-net over F128, using
(10, 10+4, 423687)-Net in Base 8 — Upper bound on s
There is no (10, 14, 423688)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 4 398048 584917 > 814 [i]